14601:
16736:
13849:
16165:
14596:{\displaystyle {\begin{aligned}\pi &=2\left(1+{\cfrac {1}{3}}+{\cfrac {1\cdot 2}{3\cdot 5}}+{\cfrac {1\cdot 2\cdot 3}{3\cdot 5\cdot 7}}+{\cfrac {1\cdot 2\cdot 3\cdot 4}{3\cdot 5\cdot 7\cdot 9}}+{\cfrac {1\cdot 2\cdot 3\cdot 4\cdot 5}{3\cdot 5\cdot 7\cdot 9\cdot 11}}+\cdots \right)\\&=2\sum _{n=0}^{\infty }{\cfrac {n!}{(2n+1)!!}}=\sum _{n=0}^{\infty }{\cfrac {2^{n+1}n!^{2}}{(2n+1)!}}=\sum _{n=0}^{\infty }{\cfrac {2^{n+1}}{{\binom {2n}{n}}(2n+1)}}\\&=2+{\frac {2}{3}}+{\frac {4}{15}}+{\frac {4}{35}}+{\frac {16}{315}}+{\frac {16}{693}}+{\frac {32}{3003}}+{\frac {32}{6435}}+{\frac {256}{109395}}+{\frac {256}{230945}}+\cdots \end{aligned}}}
5429:
16731:{\displaystyle {\begin{aligned}\pi &=6\sin ^{-1}\left({\frac {1}{2}}\right)=6\left({\frac {1}{2}}+{\frac {1}{2\cdot 3\cdot 2^{3}}}+{\frac {1\cdot 3}{2\cdot 4\cdot 5\cdot 2^{5}}}+{\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6\cdot 7\cdot 2^{7}}}+\cdots \!\right)\\&={\frac {3}{16^{0}\cdot 1}}+{\frac {6}{16^{1}\cdot 3}}+{\frac {18}{16^{2}\cdot 5}}+{\frac {60}{16^{3}\cdot 7}}+\cdots \!=\sum _{n=0}^{\infty }{\frac {3\cdot {\binom {2n}{n}}}{16^{n}(2n+1)}}\\&=3+{\frac {1}{8}}+{\frac {9}{640}}+{\frac {15}{7168}}+{\frac {35}{98304}}+{\frac {189}{2883584}}+{\frac {693}{54525952}}+{\frac {429}{167772160}}+\cdots \end{aligned}}}
4827:
2158:
1406:
2307:
calculation erroneous; due to the nature of Machin's formula, the error propagated back to the 528th decimal place, leaving only the first 527 digits correct once again. Twenty years later, Shanks expanded his calculation to 707 decimal places in April 1873. Due to this being an expansion of his previous calculation, most of the new digits were incorrect as well. Shanks was said to have calculated new digits all morning and would then spend all afternoon checking his morning's work. This was the longest expansion of
5424:{\displaystyle {\begin{aligned}\arctan x&={\frac {x}{1+x^{2}}}\sum _{k=0}^{\infty }{\frac {(2k)!!\,x^{2k}}{(2k+1)!!\,(1+x^{2})^{k}}}={\frac {x}{1+x^{2}}}+{\frac {2}{3}}{\frac {x^{3}}{(1+x^{2})^{2}}}+{\frac {2\cdot 4}{3\cdot 5}}{\frac {x^{5}}{(1+x^{2})^{3}}}+\cdots \\{\frac {\pi }{2}}&=\sum _{k=0}^{\infty }{\frac {k!}{(2k+1)!!}}=\sum _{k=0}^{\infty }{\cfrac {2^{k}k!^{2}}{(2k+1)!}}=1+{\frac {1}{3}}\left(1+{\frac {2}{5}}\left(1+{\frac {3}{7}}\left(1+\cdots \right)\right)\right)\end{aligned}}}
47:
1816:
11409:
11362:
4813:
18021:
17421:
2153:{\displaystyle {\frac {\pi }{2}}=\prod _{n=1}^{\infty }{\frac {4n^{2}}{4n^{2}-1}}=\prod _{n=1}^{\infty }\left({\frac {2n}{2n-1}}\cdot {\frac {2n}{2n+1}}\right)={\Big (}{\frac {2}{1}}\cdot {\frac {2}{3}}{\Big )}\cdot {\Big (}{\frac {4}{3}}\cdot {\frac {4}{5}}{\Big )}\cdot {\Big (}{\frac {6}{5}}\cdot {\frac {6}{7}}{\Big )}\cdot {\Big (}{\frac {8}{7}}\cdot {\frac {8}{9}}{\Big )}\cdot \;\cdots }
1400:
12348:
13087:
18286:
20194:
19434:, pp. 12, 21β22 "in 1936, a tablet was excavated some 200 miles from Babylon. ... The mentioned tablet, whose translation was partially published only in 1950, ... states that the ratio of the perimeter of a regular hexagon to the circumference of the circumscribed circle equals a number which in modern notation is given by 57/60+36/(60) ".
15678:
11197:
11009:
11888:
4499:
4513:
15932:
13405:
8058:
17714:
17114:
1129:
11117:
10523:
8403:
11163:
2600:. For one, it was known that any error would produce a value slightly high, and for the other, it was known that any error would produce a value slightly low. And hence, as long as the two series produced the same digits, there was a very high confidence that they were correct. The first 100,265 digits of
12681:
3123:, a discrepancy of nearly 2 percent. A mathematics professor who happened to be present the day the bill was brought up for consideration in the Senate, after it had passed in the House, helped to stop the passage of the bill on its second reading, after which the assembly thoroughly ridiculed it before
16940:
19097:
by Kanada
Laboratory in the University of Tokyo is the program for Microsoft Windows for runs from 16,000 to 33,550,000 digits. It can compute one million digits in 40 minutes, two million digits in 90 minutes and four million digits in 220 minutes on a Pentium 90 MHz. Super PI version 1.9 is
2306:
to 530 decimal places in
January 1853, of which the first 527 were correct (the last few likely being incorrect due to round-off errors). He subsequently expanded his calculation to 607 decimal places in April 1853, but an error introduced right at the 530th decimal place rendered the rest of his
18636:
11399:(i.e. 7) than the number of digits needed to approximate it (i.e. 6). The accuracy can be improved by using other fractions with larger numerators and denominators, but, for most such fractions, more digits are required in the approximation than correct significant figures achieved in the result.
6919:
18036:
15319:
13766:
6375:, giving about 100 digits in three steps and over a trillion digits after 20 steps. Even though the Chudnovsky series is only linearly convergent, the Chudnovsky algorithm might be faster than the iterative algorithms in practice; that depends on technological factors such as memory sizes and
3281:
based on the idea that the perimeter of any (convex) polygon inscribed in a circle is less than the circumference of the circle, which, in turn, is less than the perimeter of any circumscribed polygon. He started with inscribed and circumscribed regular hexagons, whose perimeters are readily
19997:
17657:
8808:
11357:{\displaystyle {\frac {3}{1}},{\frac {22}{7}},{\frac {333}{106}},{\frac {355}{113}},{\frac {103993}{33102}},{\frac {104348}{33215}},{\frac {208341}{66317}},{\frac {312689}{99532}},{\frac {833719}{265381}},{\frac {1146408}{364913}},{\frac {4272943}{1360120}},{\frac {5419351}{1725033}}}
6657:
6534:
4324:
4203:
1629:
9746:
15330:
10806:
11680:
4360:
4808:{\displaystyle \pi ={\sqrt {12}}\sum _{k=0}^{\infty }{\frac {(-3)^{-k}}{2k+1}}={\sqrt {12}}\sum _{k=0}^{\infty }{\frac {(-{\frac {1}{3}})^{k}}{2k+1}}={\sqrt {12}}\left({1 \over 1\cdot 3^{0}}-{1 \over 3\cdot 3^{1}}+{1 \over 5\cdot 3^{2}}-{1 \over 7\cdot 3^{3}}+\cdots \right)}
2809:
10072:
5870:
18438:
5682:
2477:
9920:
15140:
6259:
2893:. This was the world record for any type of calculation, but significantly it was performed on a home computer built by Kondo. The calculation was done between 4 May and 3 August, with the primary and secondary verifications taking 64 and 66 hours respectively.
14717:
3586:
Archimedes uses no trigonometry in this computation and the difficulty in applying the method lies in obtaining good approximations for the square roots that are involved. Trigonometry, in the form of a table of chord lengths in a circle, was probably used by
8954:
15772:
18016:{\displaystyle \pi ={\frac {1}{2^{6}}}\sum _{n=0}^{\infty }{\frac {{(-1)}^{n}}{2^{10n}}}\left(-{\frac {2^{5}}{4n+1}}-{\frac {1}{4n+3}}+{\frac {2^{8}}{10n+1}}-{\frac {2^{6}}{10n+3}}-{\frac {2^{2}}{10n+5}}-{\frac {2^{2}}{10n+7}}+{\frac {1}{10n+9}}\right)}
13102:
10795:
9446:
4079:
7880:
17416:{\displaystyle \pi ={\frac {1}{2^{6}}}\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2^{10n}}}\left(-{\frac {2^{5}}{4n+1}}-{\frac {1}{4n+3}}+{\frac {2^{8}}{10n+1}}-{\frac {2^{6}}{10n+3}}-{\frac {2^{2}}{10n+5}}-{\frac {2^{2}}{10n+7}}+{\frac {1}{10n+9}}\right)}
3980:
1395:{\displaystyle \pi ={\sqrt {12}}\sum _{k=0}^{\infty }{\frac {(-3)^{-k}}{2k+1}}={\sqrt {12}}\sum _{k=0}^{\infty }{\frac {(-{\frac {1}{3}})^{k}}{2k+1}}={\sqrt {12}}\left(1-{1 \over 3\cdot 3}+{1 \over 5\cdot 3^{2}}-{1 \over 7\cdot 3^{3}}+\cdots \right)}
14856:
10196:
13545:
9130:
3449:
11086:
3282:
determined. He then shows how to calculate the perimeters of regular polygons of twice as many sides that are inscribed and circumscribed about the same circle. This is a recursive procedure which would be described today as follows: Let
12675:
9559:
18890:
using a distributed network of several hundred computers. In 2000, after two years, the project finished computing the five trillionth (5*10), the forty trillionth, and the quadrillionth (10) bits. All three of them turned out to be 0.
13082:{\displaystyle \pi ={\cfrac {4}{1+{\cfrac {1^{2}}{3+{\cfrac {2^{2}}{5+{\cfrac {3^{2}}{7+{\cfrac {4^{2}}{9+\ddots }}}}}}}}}}=3+{\cfrac {1^{2}}{5+{\cfrac {4^{2}}{7+{\cfrac {3^{2}}{9+{\cfrac {6^{2}}{11+{\cfrac {5^{2}}{13+\ddots }}}}}}}}}}}
10354:
8242:
16031:
7874:
6790:
11123:
9834:
19673:"Add four to one hundred, multiply by eight and then add sixty-two thousand. The result is approximately the circumference of a circle of diameter twenty thousand. By this rule the relation of the circumference to diameter is given."
7789:
1063:
10326:
16773:
2238:
6417:
with 1 terabyte of main memory, which carries out 2 trillion operations per second, nearly twice as many as the computer used for the previous record (206 billion digits). The following Machin-like formulae were used for this:
3237:
There is still some debate on this passage in biblical scholarship. Many reconstructions of the basin show a wider brim (or flared lip) extending outward from the bowl itself by several inches to match the description given in
18281:{\displaystyle {\frac {\pi }{2}}=\sum _{k=0}^{\infty }{\frac {k!}{(2k+1)!!}}=\sum _{k=0}^{\infty }{\frac {2^{k}k!^{2}}{(2k+1)!}}=1+{\frac {1}{3}}\left(1+{\frac {2}{5}}\left(1+{\frac {3}{7}}\left(1+\cdots \right)\right)\right)}
3716:
18474:
6802:
8633:
20189:{\displaystyle {\overline {{\tfrac {16}{5}}-{\tfrac {4}{239}}}}-{\tfrac {1}{3}}{\overline {{\tfrac {16}{5^{3}}}-{\tfrac {4}{239^{3}}}}}+{\tfrac {1}{5}}{\overline {{\tfrac {16}{5^{5}}}-{\tfrac {4}{239^{5}}}}}-,\,\&c.=}
15151:
13597:
2539:
9619:
8477:
17495:
17080:
9202:
8686:
5523:
10620:
8179:
835:
using fewer than five decimal digits in the numerator and denominator. Zu
Chongzhi's results surpass the accuracy reached in Hellenistic mathematics, and would remain without improvement for close to a millennium.
15673:{\displaystyle {\begin{aligned}&a_{1}(x)=2/x,\\&b_{1}(x)=1,\\&a_{n}(x)=a_{n-1}(x)\,\left(1-4/x^{2}\right)+4b_{n-1}(x)/x,\\&b_{n}(x)=b_{n-1}(x)\,\left(1-4/x^{2}\right)-4a_{n-1}(x)/x,\end{aligned}}}
11004:{\displaystyle {\begin{aligned}a&={\tfrac {1}{2}}(23+4{\sqrt {34}})\\b&={\tfrac {1}{2}}(19{\sqrt {2}}+7{\sqrt {17}})\\c&=(429+304{\sqrt {2}})\\d&={\tfrac {1}{2}}(627+442{\sqrt {2}})\end{aligned}}}
11883:{\displaystyle \pi =\lim _{r\to \infty }{\frac {1}{r^{2}}}\sum _{x=-r}^{r}\;\sum _{y=-r}^{r}{\begin{cases}1&{\text{if }}{\sqrt {x^{2}+y^{2}}}\leq r\\0&{\text{if }}{\sqrt {x^{2}+y^{2}}}>r.\end{cases}}}
8550:
6547:
6424:
4214:
4093:
1464:
9636:
19058:
by Steve
Pagliarulo for Windows is faster than PiFast for runs of under 400 million digits. Version 4.5 is available on Stu's Pi Page below. Like PiFast, QuickPi can also compute other irrational numbers like
16154:
4494:{\displaystyle {\begin{aligned}\pi &\approx 768{\sqrt {2-{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2+1}}}}}}}}}}}}}}}}}}\\&\approx 3.141590463236763.\end{aligned}}}
7577:
8235:
7506:
3842:
4832:
3570:
2636:
2267:
The magnitude of such precision (152 decimal places) can be put into context by the fact that the circumference of the largest known object, the observable universe, can be calculated from its diameter
9938:
9345:
5990:
5700:
19641:
18300:
9279:
7437:
7382:
7328:
5544:
3242:
In the succeeding verses, the rim is described as "a handbreadth thick; and the brim thereof was wrought like the brim of a cup, like the flower of a lily: it received and held three thousand baths"
3091:
The so-called "Indiana Pi Bill" from 1897 has often been characterized as an attempt to "legislate the value of Pi". Rather, the bill dealt with a purported solution to the problem of geometrically "
2339:
8680:
50:
Graph showing the historical evolution of the record precision of numerical approximations to pi, measured in decimal places (depicted on a logarithmic scale; time before 1400 is not shown to scale)
7123:
19382:
A group of mathematical clay tablets from the Old
Babylonian Period, excavated at Susa in 1936, and published by E.M. Bruins in 1950, provide the information that the Babylonian approximation of
18642:
The speed of various algorithms for computing pi to n correct digits is shown below in descending order of asymptotic complexity. M(n) is the complexity of the multiplication algorithm employed.
9029:
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15335:
13854:
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4365:
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7037:
1697:
14964:
5998:
1729:
1122:
14612:
11658:
1788:
demonstrated that the perimeter of the inscribed polygon converges on the circumference twice as fast as does the perimeter of the corresponding circumscribed polygon. This was proved by
8828:
15927:{\displaystyle {\frac {\pi }{2}}=\sum _{n=0}^{\infty }\arctan {\frac {1}{F_{2n+1}}}=\arctan {\frac {1}{1}}+\arctan {\frac {1}{2}}+\arctan {\frac {1}{5}}+\arctan {\frac {1}{13}}+\cdots }
14915:
6978:
13400:{\displaystyle \pi =4\sum _{n=1}^{m}{\frac {(-1)^{n-1}}{2n-1}}+{\cfrac {2(-1)^{m}}{2m+{\cfrac {1^{2}}{2m+{\cfrac {2^{2}}{2m+{\cfrac {3^{2}}{2m+\ddots }}}}}}}}\qquad (m=1,2,3,\ldots )}
11551:
7686:
19044:. It can also work at lesser efficiency with very little memory (down to a few tens of megabytes to compute well over a billion (10) digits). This tool is a popular benchmark in the
8053:{\displaystyle 4\left(1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}\right)-{\frac {2}{10+{\frac {1^{2}}{10+{\frac {2^{2}}{10+{\frac {3^{2}}{10}}}}}}}}=3.14159\ 3^{+}}
7183:
7274:
786:(accuracy 9Β·10). He also suggested that 3.14 was a good enough approximation for practical purposes. He has also frequently been credited with a later and more accurate result, Ο β
11058:
10635:
9362:
3991:
6331:
2882:. Calculations were performed in base 2 (binary), then the result was converted to base 10 (decimal). The calculation, conversion, and verification steps took a total of 131 days.
18853:
11393:
3895:
14952:
977:
745:
14757:
10102:
18779:
18717:
20328:
13416:
11464:
9046:
11112:{\displaystyle {\frac {15261343909396942111177730086852826352374060766771618308167575028500999}{48590509502030754798379641288876701245663220023884870402810360529259}}...}
3316:
2320:
616:
The Sun is eight thousand yojanas and another two thousand yojanas in diameter. From that its peripheral circle comes to be equal to thirty thousand yojanas.
10518:{\displaystyle {\frac {\ln(2^{-30}((3+{\sqrt {5}})({\sqrt {5}}+{\sqrt {7}})({\sqrt {7}}+{\sqrt {11}})({\sqrt {11}}+3))^{12}-24)}{{\sqrt {5}}{\sqrt {7}}{\sqrt {11}}}}}
8398:{\displaystyle \left({\frac {2\cdot 4\cdot 6}{3\cdot 5\cdot 7}}\right)^{2}\left(15+{\frac {1^{2}}{30+{\frac {3^{2}}{30+{\frac {5^{2}}{30}}}}}}\right)=3.14159\ 27^{+}}
7227:
6366:
3106:, although the closest it comes to explicitly asserting one is the wording "the ratio of the diameter and circumference is as five-fourths to four", which would make
20377:
We should note that Vega's value contains an error in the 127th digit. Vega gives a 4 where there should be an [6], and all digits after that are incorrect.
16061:
12531:
11158:{\displaystyle ...{\frac {551152789881364457516133280872003443353677807669620554743{\sqrt {10005}}}{3134188302895457201473978137944378665098227220269702217081111}}}
9497:
3618:
in 1593, was derived by Viète using a closely related polygonal method, but with areas rather than perimeters of polygons whose numbers of sides are powers of two.
16089:
15764:
15721:
15701:
14749:
13819:
2864:
15943:
7797:
6703:
9772:
15741:
13799:
7203:
4087:
The record as of
December 2002 by Yasumasa Kanada of Tokyo University stood at 1,241,100,000,000 digits. The following Machin-like formulae were used for this:
7703:
3610:(1621) obtains better bounds from a pair of bounds obtained from the polygon method. Thus, more accurate results were obtained from polygons with fewer sides.
984:
612:
The Moon is handed down by memory to be eleven thousand yojanas in diameter. Its peripheral circle happens to be thirty three thousand yojanas when calculated.
16935:{\displaystyle \pi =\sum _{n=0}^{\infty }\left({\frac {4}{8n+1}}-{\frac {2}{8n+4}}-{\frac {1}{8n+5}}-{\frac {1}{8n+6}}\right)\left({\frac {1}{16}}\right)^{n}}
21245:
19357:
19638:
19251:, supposedly built so that the circle whose radius is equal to the height of the pyramid has a circumference equal to the perimeter of the base (it is 1760
10246:
6667:
These approximations have so many digits that they are no longer of any practical use, except for testing new supercomputers. Properties like the potential
477:
in particular by the mid-first millennium, to an accuracy of seven decimal places. After this, no further progress was made until the late medieval period.
2921: Γ 10)). The computation took (with three interruptions) 105 days to complete, the limitation of further expansion being primarily storage space.
18631:{\displaystyle {\frac {1}{\pi }}={\frac {1}{426880{\sqrt {10005}}}}\sum _{k=0}^{\infty }{\frac {(6k)!(13591409+545140134k)}{(3k)!(k!)^{3}(-640320)^{3k}}}}
6914:{\displaystyle \approxeq 3+{\frac {8}{60}}+{\frac {30}{60^{2}}}-{\frac {30}{74^{3}}}+{\frac {555}{55^{6}}}+{\frac {1105}{86^{8}}}=3.141592653589793237...}
3229:
can only be known approximately, so the value 3 was given as accurate enough for religious purposes. This is taken by some as the earliest assertion that
2184:
21395:
21000:
19184:
3653:
19676:
In other words, (4 + 100) Γ 8 + 62000 is the circumference of a circle with diameter 20000. This provides a value of Ο β
15314:{\displaystyle \arctan(x)=2\sum _{n=1}^{\infty }{{\frac {1}{2n-1}}{\frac {{{a}_{n}}\left(x\right)}{a_{n}^{2}\left(x\right)+b_{n}^{2}\left(x\right)}}},}
13761:{\displaystyle \pi =4\sum _{n=0}^{\infty }{\cfrac {(-1)^{n}}{2n+1}}=4\left({\frac {1}{1}}-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+-\cdots \right)}
18449:
This converges extraordinarily rapidly. Ramanujan's work is the basis for the fastest algorithms used, as of the turn of the millennium, to calculate
11663:
The total number of cells satisfying that condition thus approximates the area of the circle, which then can be used to calculate an approximation of
8556:
7055:
2896:
In
October 2011, Shigeru Kondo broke his own record by computing ten trillion (10) and fifty digits using the same method but with better hardware.
2496:
20685:
17652:{\displaystyle \pi =\sum _{k=0}^{\infty }{\frac {1}{16^{k}}}\left({\frac {4}{8k+1}}-{\frac {2}{8k+4}}-{\frac {1}{8k+5}}-{\frac {1}{8k+6}}\right).}
8803:{\displaystyle \left({\frac {\sqrt {58}}{4}}-{\frac {37{\sqrt {2}}}{33}}\right)^{-1}={\frac {66{\sqrt {2}}}{33{\sqrt {29}}-148}}=3.14159\ 263^{+}}
11416:: as points are randomly scattered inside the unit square, some fall within the unit circle. The fraction of points inside the circle approaches
3606:(when the methods are known) were made by increasing the number of sides of the polygons used in the computation. A trigonometric improvement by
20917:
9566:
8420:
21077:
16986:
9147:
20211:(among others for the same purpose, and drawn from the same Principle) I receiv'd from the Excellent Analyst, and my much Esteem'd Friend Mr.
5462:
10552:
8122:
6652:{\displaystyle {\frac {\pi }{4}}=44\arctan {\frac {1}{57}}+7\arctan {\frac {1}{239}}-12\arctan {\frac {1}{682}}+24\arctan {\frac {1}{12943}}}
6529:{\displaystyle {\frac {\pi }{4}}=12\arctan {\frac {1}{49}}+32\arctan {\frac {1}{57}}-5\arctan {\frac {1}{239}}+12\arctan {\frac {1}{110443}}}
6379:. For breaking world records, the iterative algorithms are used less commonly than the Chudnovsky algorithm since they are memory-intensive.
4319:{\displaystyle {\frac {\pi }{4}}=44\arctan {\frac {1}{57}}+7\arctan {\frac {1}{239}}-12\arctan {\frac {1}{682}}+24\arctan {\frac {1}{12943}}}
4198:{\displaystyle {\frac {\pi }{4}}=12\arctan {\frac {1}{49}}+32\arctan {\frac {1}{57}}-5\arctan {\frac {1}{239}}+12\arctan {\frac {1}{110443}}}
2943:
1624:{\displaystyle \pi /4\approx 1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots -{\frac {(-1)^{n}}{2n-1}}\pm {\frac {n^{2}+1}{4n^{3}+5n}}}
9741:{\displaystyle \left({\frac {\sqrt {253}}{4}}-{\frac {643{\sqrt {11}}}{903}}-{\frac {223}{172}}\right)^{-1}=3.14159\ 26535\ 89793\ 2387^{+}}
13781: = 1. It converges too slowly to be of practical interest. However, the power series converges much faster for smaller values of
12505:
given below involve repeated calculations of some sort, yielding closer and closer approximations with increasing numbers of calculations.
8483:
21199:
19968:
19964:
19958:
16105:
2608:
to 1 million decimal places and concluded that the task was beyond that day's technology, but would be possible in five to seven years.
20945:
916:
896:
21418:"Investigatio quarundam serierum, quae ad rationem peripheriae circuli ad diametrum vero proxime definiendam maxime sunt accommodatae"
7527:
21904:
21879:
8185:
3207:
rim. This interpretation implies a brim about 0.225 cubit (or, assuming an 18-inch "cubit", some 4 inches), or one and a third "
11592:. Squares whose center resides inside or exactly on the border of the circle can then be counted by testing whether, for each cell (
7454:
3746:
2804:{\displaystyle {\frac {1}{\pi }}=12\sum _{k=0}^{\infty }{\frac {(-1)^{k}(6k)!(13591409+545140134k)}{(3k)!(k!)^{3}640320^{3k+3/2}}}.}
19560:
10067:{\displaystyle {\frac {1}{10}}\ln \left({\frac {2^{21}}{({\sqrt{5}}-1)^{24}}}+24\right)=3.14159\ 26535\ 89793\ 23846\ 26433\ 9^{+}}
5865:{\displaystyle {\frac {1}{\pi }}=12\sum _{k=0}^{\infty }{\frac {(-1)^{k}(6k)!(13591409+545140134k)}{(3k)!(k!)^{3}640320^{3k+3/2}}}}
3520:
796:= 3.1416 (accuracy 2Β·10), although some scholars instead believe that this is due to the later (5th-century) Chinese mathematician
20369:
18433:{\displaystyle {\frac {1}{\pi }}={\frac {2{\sqrt {2}}}{9801}}\sum _{k=0}^{\infty }{\frac {(4k)!(1103+26390k)}{(k!)^{4}396^{4k}}}}
5677:{\displaystyle {\frac {1}{\pi }}={\frac {2{\sqrt {2}}}{9801}}\sum _{k=0}^{\infty }{\frac {(4k)!(1103+26390k)}{(k!)^{4}396^{4k}}}}
2593:
2472:{\displaystyle {\frac {1}{\pi }}={\frac {2{\sqrt {2}}}{9801}}\sum _{k=0}^{\infty }{\frac {(4k)!(1103+26390k)}{(k!)^{4}396^{4k}}}}
419:
achieved sixteen digits next. Early modern mathematicians reached an accuracy of 35 digits by the beginning of the 17th century (
9296:
5917:
21214:
17478:
16750:
1432:
9230:
7388:
7334:
7280:
855:
Add 4 to 100, multiply by 8 and add to 62,000. This is 'approximately' the circumference of a circle whose diameter is 20,000.
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22302:
22283:
22256:
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21811:
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21504:
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19283:
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19049:
10206:
to the integer 640320+744. This does not admit obvious generalizations in the integers, because there are only finitely many
8639:
20639:
2906:
In
October 2014, Sandon Van Ness, going by the pseudonym "houkouonchi" used y-cruncher to calculate 13.3 trillion digits of
22379:
21400:
20896:
7075:
3583:(about 60 CE) that Archimedes continued the computation in a now lost book, but then attributes an incorrect value to him.
9915:{\displaystyle {\frac {2286635172367940241408{\sqrt {2}}}{1029347477390786609545}}=3.14159\ 26535\ 89793\ 23846\ 2649^{+}}
8986:
8075:
7594:
911:
Further progress was not made for nearly a millennium, until the 14th century, when Indian mathematician and astronomer
520:). The Babylonians were aware that this was an approximation, and one Old Babylonian mathematical tablet excavated near
19622:
19375:
15135:{\displaystyle \arctan(x)=\sum _{n=0}^{\infty }{\frac {2^{2n}(n!)^{2}}{(2n+1)!}}\;{\frac {x^{2n+1}}{(1+x^{2})^{n+1}}}.}
13571:
are respectively the second and fourth continued fraction approximations to Ο. (Other representations are available at
6995:
6254:{\displaystyle y_{k+1}=(1-f(y_{k}))/(1+f(y_{k}))~,~a_{k+1}=a_{k}(1+y_{k+1})^{4}-2^{2k+3}y_{k+1}(1+y_{k+1}+y_{k+1}^{2})}
2261:
1670:
831:, which are not as accurate as his decimal result. The latter fraction is the best possible rational approximation of
22321:
16976:
time. The algorithm requires virtually no memory for the storage of an array or matrix so the one-millionth digit of
14712:{\displaystyle \pi =2+{\frac {1}{3}}\left(2+{\frac {2}{5}}\left(2+{\frac {3}{7}}\left(2+\cdots \right)\right)\right)}
3059:" of 1897, which stated "the ratio of the diameter and circumference is as five-fourths to four" (which would imply "
135:
22160:
21614:
8949:{\displaystyle {\sqrt{3^{4}+2^{4}+{\frac {1}{2+({\frac {2}{3}})^{2}}}}}={\sqrt{\frac {2143}{22}}}=3.14159\ 2652^{+}}
1708:
443:
408:, this was improved to approximations correct to what corresponds to about seven decimal digits by the 5th century.
278:
24:
20716:
18457:
16767:βreturning the hexadecimal value of the digitβwithout having to compute the intervening digits (digit extraction).
5687:
2957:
On 8 June 2022, Emma Haruka Iwao announced on the Google Cloud Blog the computation of 100 trillion (10) digits of
1071:
369:
21241:
21026:
11610:
3246:, which suggests a shape that can be encompassed with a string shorter than the total length of the brim, e.g., a
2486:
with each term in the series. His series are now the basis for the fastest algorithms currently used to calculate
22364:
22190:
21563:
19832:
Boris A. Rosenfeld & Adolf P. Youschkevitch (1981). "Ghiyath al-din
Jamshid Masud al-Kashi (or al-Kashani)".
17466:
9212:
yields the sequence 1122211125... Replacing 5 by 2 completes the symmetry without reducing the correct digits of
3575:
It is not known why
Archimedes stopped at a 96-sided polygon; it only takes patience to extend the computations.
3154:
2913:
In November 2016, Peter Trueb and his sponsors computed on y-cruncher and fully verified 22.4 trillion digits of
20874:
19001:
by Alexander Yee is the program which every world record holder since Shigeru Kondo in 2010 has used to compute
3263:
2252:
used a similar identity to calculate 127 digits (of which 112 were correct). In 1789, the Slovene mathematician
114:
19538:
Damini, D.B.; Abhishek, Dhar (2020). "How Archimedes showed that Ο is approximately equal to 22/7". p. 8.
19367:
18946:
18907:
14864:
12522:
6943:
2596:
in Washington, D.C. In 1961, Shanks and his team used two different power series for calculating the digits of
21340:
Unpublished work by Newton (1684), later independently discovered by others, and popularized by Euler (1755).
11503:
10790:{\displaystyle u=(a+{\sqrt {a^{2}-1}})^{2}(b+{\sqrt {b^{2}-1}})^{2}(c+{\sqrt {c^{2}-1}})(d+{\sqrt {d^{2}-1}})}
9441:{\displaystyle \left({\frac {\sqrt {163}}{6}}-{\frac {181}{\sqrt {10005}}}\right)^{-1}=3.14159\ 26535\ 89^{+}}
7649:
4074:{\displaystyle {\frac {\pi }{4}}=12\arctan {\frac {1}{18}}+8\arctan {\frac {1}{57}}-5\arctan {\frac {1}{239}}}
3874:. Machin's particular formula was used well into the computer era for calculating record numbers of digits of
340:
19954:
7140:
3150:
232:
7237:
438:, who calculated 527 decimals correctly in 1853. Since the middle of the 20th century, the approximation of
130:
22072:
19122:
18659:
17451:
11030:
5896:
3975:{\displaystyle {\frac {\pi }{4}}=6\arctan {\frac {1}{8}}+2\arctan {\frac {1}{57}}+\arctan {\frac {1}{239}}}
3016:(relative error of about 8Β·10). In Chinese mathematics, the fractions 22/7 and 355/113 are known as YuelΓΌ (
449:). On June 28, 2024, the current record was established by the StorageReview Lab team with Alexander Yee's
161:
20475:
Contributions to Mathematics: Comprising Chiefly the Rectification of the Circle to 607 Places of Decimals
6267:
3243:
3216:
512:
to 3, sufficient for the architectural projects of the time (notably also reflected in the description of
151:
22039:
Plouffe, Simon (2009). "On the computation of the n^th decimal digit of various transcendental numbers".
18988:
by Fabrice Bellard is the program used by himself to compute world record number of digits of pi in 2009.
18798:
17459:
14851:{\displaystyle {\frac {\pi }{2^{k+1}}}=\arctan {\frac {\sqrt {2-a_{k-1}}}{a_{k}}},\qquad \qquad k\geq 2,}
11475:
is drawn with its center at the point (0, 0), any point whose distance from the origin is less than
11181:
10191:{\displaystyle {\frac {\ln(640320^{3}+744)}{\sqrt {163}}}=3.14159\ 26535\ 89793\ 23846\ 26433\ 83279^{+}}
2992:
can be approximated by using fractions for ease of calculation. The most notable such approximations are
21052:
16980:
can be computed using a pocket calculator. However, it would be quite tedious and impractical to do so.
16091:
are much slower in convergence because of set of arctangent functions that are involved in computation.
11371:
4347:
2899:
In December 2013, Kondo broke his own record for a second time when he computed 12.1 trillion digits of
187:
20682:
20304:
19763:
19452:
19275:
14920:
938:
692:
546:
542:
22146:
17682: β 1 digits. Bailey's website contains the derivation as well as implementations in various
3239:
3212:
182:
20807:
Treub, Peter (30 November 2016). "Digit Statistics of the First 22.4 Trillion Decimal Digits of Pi".
19856:
19799:
19112:
19027:
19021:
in 2003. According to its author, it can compute one million digits in 3.5 seconds on a 2.4 GHz
18950:
18733:
13540:{\displaystyle {\frac {2}{2m+{\frac {1^{2}}{2m+{\frac {2^{2}}{2m}}}}}}=4{\frac {m^{2}+1}{4m^{3}+5m}}}
9125:{\displaystyle {\frac {63}{25}}\times {\frac {17+15{\sqrt {5}}}{7+15{\sqrt {5}}}}=3.14159\ 26538^{+}}
7206:
2545:
21088:
20498:
19011:-cruncher can also be used to calculate other constants and holds world records for several of them.
18669:
11777:
7791:
20545:
20510:
19711:
19707:
19693:
3885:
and his team used the following Machin-like formula in 1961 to compute the first 100,000 digits of
3444:{\displaystyle P_{2n}={\frac {2p_{n}P_{n}}{p_{n}+P_{n}}},\quad \quad p_{2n}={\sqrt {p_{n}P_{2n}}}.}
3099:
21980:
Hwang Chien-Lih (2005), "An elementary derivation of Euler's series for the arctangent function",
21437:
Hwang Chien-Lih (2005), "An elementary derivation of Euler's series for the arctangent function",
19667:
chaturadhikam ΕatamaαΉ£αΉaguαΉam dvΔΕaαΉ£αΉistathΔ sahasrΔαΉΔm ayutadvayaviαΉ£kambhasyΔsanno vrΜ₯ttapariαΉahaαΈ₯
18859:
13093:
11430:
1645:
932:
416:
207:
22359:
20974:
20233:
18973:
may have better performance than general-purpose mathematical software. They typically implement
18923:
11929:. Starting at 0, add 1 for each cell whose distance to the origin (0,0) is less than or equal to
10203:
9452:
This is obtained from the Chudnovsky series (truncate the series (1.4) at the first term and let
4504:
912:
689:, the first known approximation accurate to three decimal places (accuracy 2Β·10). It is equal to
550:
427:), surpassing the accuracy required for any conceivable application outside of pure mathematics.
412:
202:
21631:
19155:
2939:
In January 2020, Timothy Mullican announced the computation of 50 trillion digits over 303 days.
567:β 3.16 (accurate to 0.6 percent) by calculating the area of a circle via approximation with the
22374:
19904:
17691:
17455:
13589:
6677:
will always depend on the infinite string of digits on the end, not on any finite computation.
5900:
2574:
had made a mistake in the 528th decimal place, and that all succeeding digits were incorrect.
2249:
899:) has argued that the word means not only that this is an approximation, but that the value is
804:
to be between 3.1415926 and 3.1415927, which was correct to seven decimal places. He also gave
505:
397:
222:
21708:
Borwein, J.; Borwein, P.; Dilcher, K. (1989). "Pi, Euler numbers, and asymptotic expansions".
21417:
20701:
19848:
12670:{\displaystyle \pi ={3+{\cfrac {1^{2}}{6+{\cfrac {3^{2}}{6+{\cfrac {5^{2}}{6+\ddots \,}}}}}}}}
9554:{\displaystyle {\frac {2510613731736{\sqrt {2}}}{1130173253125}}=3.14159\ 26535\ 89793\ 9^{+}}
20950:
20853:
19791:
19612:
19361:
19248:
18931:
17667:
11024:
10218:
7212:
6336:
3124:
2933:
2567:
1634:
It is not known how he came up with this correction. Using this he found an approximation of
386:
21673:
Dutka, J. (1982). "Wallis's product, Brouncker's continued fraction, and Leibniz's series".
20540:
16026:{\displaystyle {\frac {\pi }{4}}=\sum _{k\geq 2}\arctan {\frac {\sqrt {2-a_{k-1}}}{a_{k}}},}
15145:
Alternatively, the following simple expansion series of the arctangent function can be used
9216:, while inserting a central decimal point remarkably fixes the accompanying magnitude at 10.
7869:{\displaystyle {\sqrt {2}}+{\sqrt {3}}+{\frac {{\sqrt {2}}-{\sqrt {3}}}{68}}=3.14159\ 0^{+}}
6785:{\displaystyle 3+{\frac {8}{60}}+{\frac {29}{60^{2}}}+{\frac {44}{60^{3}}}=3.14159\ 259^{+}}
4351:
3611:
2260:'s formula to calculate the first 140 digits, of which the first 126 were correct. In 1841,
1755:
22266:
22217:
21644:
20742:
20479:
20428:
19605:
18723:
18465:
17705:
17683:
17440:
16741:
with a convergence such that each additional five terms yields at least three more digits.
16039:
12352:
9829:{\displaystyle {\frac {3949122332{\sqrt {2}}}{1777729635}}=3.14159\ 26535\ 89793\ 2382^{+}}
6376:
5876:
5446:
2815:
924:
861:
848:
524:
in 1936 (dated to between the 19th and 17th centuries BCE) gives a better approximation of
362:
19476:
16074:
15749:
15706:
15686:
14734:
13804:
11556:
Mathematical "graph paper" is formed by imagining a 1Γ1 square centered around each cell (
11424:
Pi can be obtained from a circle if its radius and area are known using the relationship:
11188:
relative to the size of their denominators. Here is a list of the first thirteen of these:
3625:
by this method was carried out by Grienberger in 1630 who calculated 39 decimal places of
3136:
8:
18942:
18789:
18442:
17436:
14606:
with a convergence such that each additional 10 terms yields at least three more digits.
13822:
11482:
7784:{\displaystyle \left(2-{\frac {\sqrt {2{\sqrt {2}}-2}}{2^{2}}}\right)^{2}=3.14159\ 6^{+}}
7230:
6368:
3870:
3638:
3615:
3607:
3588:
3177:
3162:
3092:
2823:
2612:
2326:
2321:
Chronology of computation of Ο Β§ The age of electronic computers (from 1949 onwards)
2179:
2167:
1785:
1747:
1653:
1058:{\displaystyle \pi =4\left(1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots \right)}
892:
755:
576:
517:
513:
474:
405:
382:
322:
257:
247:
217:
22221:
21648:
21429:
21405:
21218:
20828:
20473:
20432:
19568:
19399:
22040:
21997:
21962:
21690:
21510:
21482:
21454:
20808:
20621:
20522:
20454:
20285:
20277:
20226:
s Number, or that in Art. 64.38. may be Examin'd with all desireable Ease and Dispatch.
20219:
19539:
19520:
19340:
19160:
18461:
15726:
13784:
12514:
11173:
7188:
5691:
3576:
2831:
2311:
until the advent of the electronic digital computer three-quarters of a century later.
1789:
1773:
with a 2-gon. He was so proud of this accomplishment that he had them inscribed on his
1762:
420:
212:
22330:
21266:
10321:{\displaystyle {\frac {\ln(5280^{3}(236674+30303{\sqrt {61}})^{3}+744)}{\sqrt {427}}}}
3310:
sides that are inscribed and circumscribed about the same circle, respectively. Then,
22336:
22317:
22298:
22279:
22252:
22015:
22001:
21954:
21849:
21807:
21773:
21694:
21589:
21514:
21500:
21458:
21170:
20526:
20446:
20361:
20353:
20343:
20289:
19767:
19738:
19618:
19596:
19524:
19512:
19456:
19371:
19279:
19236:
19018:
18911:
6399:
3734:
2233:{\textstyle {\tfrac {1}{4}}\pi =4\operatorname {arccot} 5-\operatorname {arccot} 239}
1797:
1649:
900:
470:
37:
for other aspects of the evolution of our knowledge about mathematical properties of
22243:
22202:
21353:
21001:"StorageReview Lab Breaks Pi Calculation World Record with Over 202 Trillion Digits"
19185:"StorageReview Lab Breaks Pi Calculation World Record with Over 202 Trillion Digits"
6693:
can be approximated to eight (decimal) significant figures with the number 3;8,29,44
1664:
digits in 1424, and translated this into 16 decimal digits after the decimal point:
442:
has been the task of electronic digital computers (for a comprehensive account, see
22225:
21989:
21946:
21725:
21721:
21717:
21682:
21652:
21492:
21446:
21160:
21126:
21122:
20613:
20568:
20514:
20483:
20458:
20436:
20419:
20269:
19883:
19591:
19502:
19444:
19165:
19076:
19035:
18785:
16068:
14722:
13834:
12496:
The circle problem: number of points (x,y) in square lattice with x^2 + y^2 <= n
6407:
5904:
5438:
1431:
terms. Each subsequent subplot magnifies the shaded area horizontally by 10 times.
920:
104:
22230:
21919:
21347:. Vol. 1 (2 ed.). Cambridge University Press. pp. 215β216, 219β220.
20256:
Tweddle, Ian (1991). "John Machin and Robert Simson on Inverse-tangent Series for
19871:
19306:
6410:
stood at 1.24 trillion digits, which were computed in September 2002 on a 64-node
3711:{\displaystyle {\frac {\pi }{4}}=4\arctan {\frac {1}{5}}-\arctan {\frac {1}{239}}}
2490:. Evaluating the first term alone yields a value correct to seven decimal places:
227:
22262:
21636:
21618:
20689:
20647:
19645:
19601:
19495:
Vidyottama Sanatana: International Journal of Hindu Science and Religious Studies
19224:
17701:
17486:
17097:
8965:
6403:
3086:
3056:
2875:
2839:
2819:
1735:
840:
355:
332:
301:
284:
262:
22018:
21141:
20946:"Even more pi in the sky: Calculating 100 trillion digits of pi on Google Cloud"
13572:
8628:{\displaystyle 3+{\frac {{\sqrt {2}}{\sqrt{2}}{\sqrt{2}}}{10}}=3.14159\ 268^{+}}
6660:
4329:
1405:
891:"approaching") gave the circumference of a circle. His 15th-century commentator
22189:
21770:
Mathematics by Experiment: Plausible Reasoning in the 21st Century, 2nd Edition
21581:
21413:
21391:
20760:
20395:
19888:
19344:
10229:
10207:
7059:
3738:
3184:
3003:
2843:
2604:
were published in 1962. The authors outlined what would be needed to calculate
2571:
2299:
1808:
435:
400:
reached an accuracy within 0.04% of the true value before the beginning of the
242:
21993:
21496:
21450:
19813:
Gupta, R. C. (1992). "On the remainder term in the MadhavaβLeibniz's series".
19363:
Athletics and Mathematics in Archaic Corinth: The Origins of the Greek Stadion
5879:, the fastest algorithms used, as of the turn of the millennium, to calculate
22353:
22239:
22198:
22194:
21958:
21656:
21379:
21174:
20593:
20450:
20390:
20365:
19929:
19516:
19507:
18974:
17474:
17470:
15703:
with even more rapid convergence. Convergence in this arctangent formula for
11935:. When finished, divide the sum, representing the area of a circle of radius
11016:
9208:
This curious approximation follows the observation that the 193rd power of 1/
8814:
This is the case that cannot be obtained from Ramanujan's approximation (22).
6668:
6414:
6411:
5908:
3882:
3855:
3722:
2620:
2589:
2534:{\displaystyle \pi \approx {\frac {9801}{2206{\sqrt {2}}}}\approx 3.14159273}
2277:
2171:
627:
601:
481:
327:
109:
31:
19582:
Lam, Lay Yong; Ang, Tian Se (1986), "Circle measurements in ancient China",
19449:
The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook
19099:
11165:- improved inverse of sum of the first nineteen terms of Chudnovsky series.
3102:
in the U.S., and has been claimed to imply a number of different values for
22125:
21730:
21404:(in Latin). Academiae Imperialis Scientiarium Petropolitanae. p. 318.
21375:
21194:
21165:
20922:
20518:
19045:
18290:
14954:. Approximations can be made by using, for example, the rapidly convergent
11020:
10211:
6537:
4818:
3142:
3067:
1458:
He also improved the formula based on arctan(1) by including a correction:
19665:
19657:
19229:
The Pyramids: The Mystery, Culture, and Science of Egypt's Great Monuments
9614:{\displaystyle {\frac {165707065}{52746197}}=3.14159\ 26535\ 89793\ 4^{+}}
8472:{\displaystyle {\frac {{\sqrt {2669}}-{\sqrt {547}}}{9}}=3.14159\ 269^{+}}
885:
600:(500 BCE β 300 CE) offers an approximation of 3, in the ratios offered in
22369:
22164:
22060:
21894:
21869:
21608:
20782:
20597:
20305:"DetΓ©rmination de la demi-circonfΓ©rence d'un cercle dont le diameter est
20213:
19305:. Discussions in Egyptology. Vol. 20. pp. 25β34. Archived from
19232:
19169:
17671:
17075:{\displaystyle \pi +3=\sum _{n=1}^{\infty }{\frac {n2^{n}n!^{2}}{(2n)!}}}
16758:
10333:
9197:{\displaystyle {\sqrt{\frac {10^{100}}{11222.11122}}}=3.14159\ 26536^{+}}
7067:
7043:
3644:
3208:
2968:
On 14 March 2024, Jordan Ranous, Kevin OβBrien and Brian Beeler computed
2556:
2257:
2163:
1804:
1661:
797:
748:
631:
597:
499:
252:
237:
192:
166:
58:
20281:
19831:
18981:
to facilitate extremely long-running and memory-expensive computations.
18867:
Sublinear convergence. Five billion terms for 10 correct decimal places
11184:. These approximations are the best possible rational approximations of
5518:{\displaystyle {\pi }=20\arctan {\frac {1}{7}}+8\arctan {\frac {3}{79}}}
2871:
to roughly 2.6 trillion digits in approximately 73 hours and 36 minutes.
2291:, the shortest unit of length expected to be directly measurable) using
22102:
21966:
21934:
21843:
21801:
21686:
21546:
21481:. Springer Proceedings in Mathematics & Statistics. Vol. 313.
21386:. Vol. 4, 1674β1684. Cambridge University Press. pp. 526β653.
20625:
20273:
19989:, which may very much facilitate the Practice; as for instance, in the
19491:"On The Value Implied in the Data Referred To in the MahΔbhΔrata for Ο"
19085:. The software may be obtained from the Pi-Hacks Yahoo! forum, or from
18992:
11093:
15261343909396942111177730086852826352374060766771618308167575028500999
10615:{\displaystyle {\frac {\ln {\big (}(2u)^{6}+24{\big )}}{\sqrt {3502}}}}
8174:{\displaystyle {\frac {{\sqrt {883}}-{\sqrt {21}}}{8}}=3.14159\ 25^{+}}
3222:
3158:
3051:
Of some notability are legal or historical texts purportedly "defining
2962:
2886:
2846:, a 64-node supercomputer with 1 terabyte of main memory, to calculate
2592:(no relation to the aforementioned William Shanks) and his team at the
2273:
2253:
2175:
1774:
779:
by inscribing a 96-gon and 192-gon; the average of these two values is
641:
450:
424:
401:
46:
21529:
11395:
is the only fraction in this sequence that gives more exact digits of
2834:(128 nodes) using another variation of Ramanujan's infinite series of
2551:
From the mid-20th century onwards, all improvements in calculation of
415:
developed approximations correct to eleven and then thirteen digits.
22023:
20441:
20414:
19022:
8961:
7515:
7063:
7054:, and that this is responsible for some of Plato's confidence in the
5535:
2560:
1734:
He achieved this level of accuracy by calculating the perimeter of a
844:
197:
22045:
21950:
20617:
20572:
19611:
Berggren, J. L.; Borwein, Jonathan M.; Borwein, Peter, eds. (2004).
19490:
11408:
11096:
48590509502030754798379641288876701245663220023884870402810360529259
8545:{\displaystyle {\frac {16}{5{\sqrt{2}}{\sqrt{2}}}}=3.14159\ 260^{+}}
7058:
of mathematical geometryβand Plato's repeated discussion of special
22331:
Berggren, Lennart; Borwein, Jonathan M.; Borwein, Peter B. (2004).
21487:
20813:
20391:"What kind of accuracy could one get with Pi to 40 decimal places?"
19544:
19093:
18902:
Over the years, several programs have been written for calculating
13821:
arises as the sum of small angles with rational tangents, known as
3878:, but more recently other similar formulae have been used as well.
3196:
2623:
22158:
20717:"Short Sharp Science: Epic pi quest sets 10 trillion digit record"
20669:
20357:
19846:
19789:
16149:{\displaystyle \sin \left({\frac {\pi }{6}}\right)={\frac {1}{2}}}
11667:. Closer approximations can be produced by using larger values of
1409:
Comparison of the convergence of two Madhava series (the one with
22335:(3rd ed.). New York: Springer Science + Business Media LLC.
19127:
15766:
can also be expressed by infinite sum of arctangent functions as
12525:
representations generated by a simple rule, including these two.
11581:
9836:- improved inverse of sum of first two terms of Ramanujan series.
1746:
In the second half of the 16th century, the French mathematician
758:
676:
669:
568:
22080:
21899:"Sequence A002486 (Denominators of convergents to Pi)"
20918:"Swiss researchers calculate pi to new record of 62.8tn figures"
19420:. MΓ©moires de la Mission archΓ©ologique en Iran. Vol. XXXIV.
19335:"There has been concern over the apparent biblical statement of
19117:
19002:
11019:. Similar to the previous two, but this time is a quotient of a
7572:{\displaystyle {\frac {9}{5}}+{\sqrt {\frac {9}{5}}}=3.1416^{+}}
3199:, ca. 150 CE) by saying that the diameter was measured from the
3043:
3027:
2626:
using the following variation of Ramanujan's infinite series of
805:
22203:"On the Rapid Computation of Various Polylogarithmic Constants"
19252:
18978:
13772:
13096:
can be expressed as generalized continued fraction as follows.
8230:{\displaystyle {\frac {9801}{2206{\sqrt {2}}}}=3.14159\ 27^{+}}
6926:
In addition, the following expressions can be used to estimate
6923:
This aproximation shows us the exact of first 18 digits of Pi.
3726:
3251:
3247:
3192:
3173:
2925:
2878:
used a home computer to compute 2.7 trillion decimal digits of
473:
were accurate to two decimal places; this was improved upon in
306:
21874:"Sequence A002485 (Numerators of convergents to Pi)"
21370:. Mathematical Association of America. 2014. pp. 109β118.
20336:
Jurij baron Vega in njegov Δas: Zbornik ob 250-letnici rojstva
19757:
7501:{\displaystyle {\sqrt {7+{\sqrt {6+{\sqrt {5}}}}}}=3.1416^{+}}
3837:{\displaystyle (5+i)^{4}\cdot (239-i)=2^{2}\cdot 13^{4}(1+i).}
549:, c. 1600 BCE, although stated to be a copy of an older,
22314:
Pyramid: Beyond Imagination. Inside the Great Pyramid of Giza
21844:
Berggren, Lennart; Borwein, Jonathan; Borwein, Peter (2003).
21802:
Berggren, Lennart; Borwein, Jonathan; Borwein, Peter (2003).
18958:
18883:
17687:
14955:
11151:
3134188302895457201473978137944378665098227220269702217081111
7047:
6686:
5453:
3565:{\displaystyle 3{\frac {10}{71}}<\pi <3{\frac {1}{7}}.}
3166:
2287:
1702:
which gives 16 correct digits for Ο after the decimal point:
22297:(New ed., London : Penguin ed.). London: Penguin.
21215:"Ancient Creation Stories told by the Numbers: Solomon's Pi"
20875:"Calculating Pi: My attempt at breaking the Pi World Record"
2264:
calculated 208 digits, of which the first 152 were correct.
883:= 3.1416, Aryabhata stated that his result "approximately" (
22295:
The Crest of the Peacock: Non-European Roots of Mathematics
21898:
21873:
19629:. See in particular pp. 333β334 (pp. 28β29 of the reprint).
18954:
18026:
Other formulae that have been used to compute estimates of
17085:
The calculation speed of Plouffe's formula was improved to
16945:
In 1996, Simon Plouffe derived an algorithm to extract the
12495:
11876:
4340:
Other formulae that have been used to compute estimates of
521:
411:
Further progress was not made until the 14th century, when
21113:
Hallerberg, Arthur E. (1977). "Indiana's Squared Circle".
21087:. University of Illinois Board of Trustees. Archived from
12347:
9922:- inverse of sum of first three terms of Ramanujan series.
9340:{\displaystyle {\sqrt{8769956796}}=3.14159\ 26535\ 89^{+}}
8968:
appeared to him in a dream and told him the true value of
5985:{\displaystyle y_{0}={\sqrt {2}}-1,\ a_{0}=6-4{\sqrt {2}}}
1438:
He used the first 21 terms to compute an approximation of
21239:
18886:
was a project to compute three specific binary digits of
17100:, who derived an alternative formula (albeit only in base
16961:
10 digit), and which can do so with an improved speed of
11140:
551152789881364457516133280872003443353677807669620554743
10336:. Among negative discriminants with class number 2, this
9752:
This is the approximation (22) in Ramanujan's paper with
1416:
in dark blue) and several historical infinite series for
20897:"Die FH GraubΓΌnden kennt Pi am genauesten β Weltrekord!"
19930:"An Improvement of Archimedes Method of Approximating Ο"
19025:. PiFast can also compute other irrational numbers like
12517:
representation , which displays no discernible pattern,
9561:- inverse of sum of first two terms of Ramanujan series.
9274:{\displaystyle {\sqrt{888582403}}=3.14159\ 26535\ 8^{+}}
7432:{\displaystyle 3+{\frac {\sqrt {2}}{10}}=3.14142\ 1^{+}}
7377:{\displaystyle {\sqrt {51}}-{\sqrt {16}}=3.14142\ 8^{+}}
7323:{\displaystyle {\sqrt {63}}-{\sqrt {23}}=3.14142\ 2^{+}}
7050:
knew this expression, that he believed it to be exactly
5895:
are typically computed with iterative formulae like the
2830:
to over 200 billion decimal places on the supercomputer
2814:
Records since then have all been accomplished using the
22159:
Takahashi, Daisuke; Kanada, Yasumasa (10 August 2010).
17662:
This formula permits one to fairly readily compute the
14398:
14373:
14315:
14277:
14216:
14201:
14116:
14080:
14043:
14013:
13982:
13958:
13933:
13915:
13896:
13884:
13662:
13634:
13326:
13307:
13290:
13271:
13254:
13235:
13218:
13187:
13040:
13021:
13007:
12988:
12974:
12955:
12941:
12922:
12908:
12889:
12838:
12819:
12805:
12786:
12772:
12753:
12739:
12720:
12706:
12694:
12636:
12617:
12603:
12584:
12570:
12551:
9138:
accurate to ten digits (or eleven significant figures):
8675:{\displaystyle {\frac {99733}{31746}}=3.14159\ 264^{+}}
6672:
5296:
5264:
2850:
to roughly 1.24 trillion digits in around 600 hours (25
580:(c. 6th century BCE) use a fractional approximation of
498:= 3.142857 (about 0.04% too high) from as early as the
389:
67:
21893:
21868:
20759:
Yee, Alexander J.; Kondo, Shigeru (28 December 2013).
20588:
20586:
20584:
20582:
20580:
20143:
20121:
20106:
20078:
20056:
20041:
20020:
20005:
19610:
16763:
math, the formula can compute any particular digit of
14401:
14376:
14318:
14280:
14219:
14204:
14119:
14083:
14046:
14016:
13985:
13961:
13936:
13918:
13899:
13887:
13665:
13637:
13329:
13310:
13293:
13274:
13257:
13238:
13221:
13190:
13043:
13024:
13010:
12991:
12977:
12958:
12944:
12925:
12911:
12892:
12841:
12822:
12808:
12789:
12775:
12756:
12742:
12723:
12709:
12697:
12639:
12620:
12606:
12587:
12573:
12554:
12359:
The 12 cells (0, Β±5), (Β±5, 0), (Β±3, Β±4), (Β±4, Β±3) are
10964:
10873:
10825:
7445:
accurate to four digits (or five significant figures):
5299:
5267:
3180:. Among the many explanations and comments are these:
2588:
decimal places was computed by Maryland mathematician
2189:
2187:
21745:
Lange, L. (1999). "An elegant continued fraction for
21707:
21564:"What can you do with a supercomputer? β ExtremeTech"
21345:
Series and Products in the Development of Mathematics
20741:
Yee, Alexander J.; Kondo, Shigeru (22 October 2011).
20340:
Baron Jurij Vega and His Times: Celebrating 250 Years
20000:
19937:
International Journal of Pure and Applied Mathematics
19758:
George E. Andrews, Ranjan Roy; Richard Askey (1999).
19400:"Quelques textes mathΓ©matiques de la Mission de Suse"
19272:
Corinna Architecture and Mathematics in Ancient Egypt
18801:
18736:
18672:
18477:
18303:
18039:
17717:
17498:
17435:
were developed and published by Indian mathematician
17117:
16989:
16776:
16168:
16108:
16077:
16042:
15946:
15775:
15752:
15729:
15709:
15689:
15333:
15154:
14967:
14923:
14867:
14760:
14737:
14615:
13852:
13807:
13787:
13600:
13419:
13105:
12684:
12534:
11683:
11613:
11506:
11433:
11374:
11200:
11126:
11089:
11033:
10809:
10638:
10555:
10357:
10249:
10105:
9941:
9855:
9775:
9639:
9569:
9500:
9365:
9299:
9233:
9150:
9049:
8989:
8831:
8689:
8642:
8559:
8486:
8423:
8245:
8188:
8125:
8078:
7883:
7800:
7706:
7652:
7597:
7530:
7518:, accurate to 4 digits (or five significant figures):
7457:
7391:
7337:
7283:
7240:
7215:
7191:
7143:
7118:{\displaystyle {\sqrt {15}}-{\sqrt {3}}+1=3.1409^{+}}
7078:
6998:
6946:
6805:
6706:
6550:
6427:
6339:
6270:
6001:
5920:
5703:
5547:
5465:
4830:
4516:
4363:
4217:
4096:
3994:
3898:
3749:
3656:
3523:
3319:
3277:, created the first algorithm for the calculation of
2867:
more than doubled the previous record by calculating
2639:
2499:
2342:
1819:
1792:
in 1654. Snellius was able to obtain seven digits of
1711:
1673:
1467:
1132:
1074:
987:
941:
695:
22013:
21767:
9024:{\displaystyle {\sqrt{28658146}}=3.14159\ 26538^{+}}
3643:
For fast calculations, one may use formulae such as
3268:
3257:
2863:
In August 2009, a Japanese supercomputer called the
2577:
In the early years of the computer, an expansion of
2329:
found several rapidly converging infinite series of
19:
This page is about the history of approximations of
22311:
21588:. Mathematical Association of America. p. 92.
20975:"Limping to a new Pi Record of 105 Trillion Digits"
20577:
12355:
graph. The cells (Β±3, Β±4) and (Β±4, Β±3) are labeled.
3203:rim while the circumference was measured along the
2885:In August 2010, Shigeru Kondo used Alexander Yee's
21027:"What is the Best Fractional Representation of Pi"
20188:
19732:
18847:
18773:
18711:
18630:
18432:
18280:
18015:
17651:
17462:, which was invented in 1976, has also been used.
17415:
17074:
16934:
16730:
16148:
16099:Observing an equilateral triangle and noting that
16083:
16055:
16025:
15926:
15758:
15735:
15715:
15695:
15672:
15313:
15134:
14946:
14909:
14850:
14743:
14711:
14595:
13813:
13793:
13760:
13539:
13399:
13081:
12669:
11882:
11652:
11545:
11458:
11387:
11356:
11157:
11111:
11052:
11003:
10789:
10614:
10517:
10320:
10190:
10066:
9914:
9828:
9740:
9613:
9553:
9440:
9339:
9273:
9196:
9124:
9023:
8948:
8802:
8674:
8627:
8544:
8471:
8397:
8229:
8173:
8111:{\displaystyle {\frac {355}{113}}=3.14159\ 29^{+}}
8110:
8052:
7868:
7783:
7680:
7638:{\displaystyle {\frac {7^{7}}{4^{9}}}=3.14156^{+}}
7637:
7571:
7500:
7431:
7376:
7322:
7268:
7221:
7197:
7177:
7117:
7031:
6972:
6913:
6784:
6651:
6528:
6360:
6325:
6253:
5984:
5864:
5676:
5517:
5423:
4807:
4493:
4318:
4197:
4073:
3974:
3836:
3710:
3564:
3443:
2803:
2533:
2471:
2232:
2152:
1723:
1691:
1623:
1394:
1116:
1057:
971:
739:
22292:
20499:"V. On the extension of the numerical value of Ο"
20408:
20406:
19927:
19639:How Aryabhata got the earth's circumference right
16520:
16389:
10529:This is derived from Ramanujan's class invariant
10078:This is derived from Ramanujan's class invariant
7032:{\displaystyle {\sqrt {2}}+{\sqrt {3}}=3.146^{+}}
6402:. A former calculation record (December 2002) by
2975:On 28 June 2024, the StorageReview Team computed
2482:which computes a further eight decimal places of
2138:
2108:
2098:
2068:
2058:
2028:
2018:
1988:
1750:discovered an infinite product that converged on
22351:
20854:"The Pi Record Returns to the Personal Computer"
19790:J J O'Connor and E F Robertson (November 2000).
19735:Indian Mathematics and Astronomy: Some Landmarks
11691:
2924:In March 2019, Emma Haruka Iwao, an employee at
1692:{\displaystyle 2\pi \approx 6.2831853071795864,}
21979:
21932:
21436:
19558:
16757:was discovered in 1995 by Simon Plouffe. Using
12371:is calculated to be approximately 3.24 because
6680:
21422:Nova Acta Academiae Scientiarum Petropolitinae
20403:
20317:Nova Acta Academiae Scientiarum Petropolitanae
19537:
19052:. PiFast 4.3 is available from Gourdon's page.
19017:by Xavier Gourdon was the fastest program for
12501:Similarly, the more complex approximations of
11893:In other words, begin by choosing a value for
3733:). This formula is most easily verified using
2979:to 202 trillion digits, also using y-cruncher.
2972:to 105 trillion digits, also using y-cruncher.
2566:In 1944β45, D. F. Ferguson, with the aid of a
1769:1600) computed the first 35 decimal places of
1724:{\displaystyle \pi \approx 3.1415926535897932}
22278:(6th ed.). Saunders College Publishing.
22276:An Introduction to the History of Mathematics
21472:
21470:
21468:
21140:Tsaban, Boaz; Garber, David (February 1998).
18894:
16575:
16557:
14424:
14406:
11674:Mathematically, this formula can be written:
10599:
10567:
5441:, the product of the positive integers up to
3454:Archimedes uses this to successively compute
3304:denote the perimeters of regular polygons of
2944:University of Applied Sciences of the Grisons
2932:) trillion digits of pi using y-cruncher and
1656:, correctly computed the fractional part of 2
1117:{\displaystyle \pi =6\arctan(1/{\sqrt {3}}):}
672:(accuracies of 2Β·10 and 4Β·10, respectively).
538:= 3.125, about 0.528% below the exact value.
363:
21312:
21310:
21240:O'Connor, J J; E F Robertson (August 2001).
21139:
20592:
19973:There are various other ways of finding the
19928:Chakrabarti, Gopal; Hudson, Richard (2003).
19847:J J O'Connor and E F Robertson (July 1999).
19632:
19415:
19149:
19147:
19145:
19143:
18464:found an even faster-converging series (the
11653:{\displaystyle {\sqrt {x^{2}+y^{2}}}\leq r.}
8237:- inverse of first term of Ramanujan series.
6689:60 was used for calculations. In this base,
3055:" to have some rational value, such as the "
21200:The Ancient Tradition of Geometric Problems
20754:
20752:
19993:, the Diameter is to Circumference as 1 to
19785:
19783:
19328:
16744:
10630:is a product of four simple quartic units,
4335:
3985:and they used another Machin-like formula,
2988:Depending on the purpose of a calculation,
2983:
2946:announced completion of the computation of
1638:to 13 decimal places of accuracy when
502:. This claim has been met with skepticism.
21538:
21465:
21112:
20776:
20774:
20772:
20770:
20533:
20503:Proceedings of the Royal Society of London
19164:. Vol. 102, no. 5. p. 342.
17678:, without having to compute the preceding
15068:
14751:involving arctangent function is given by
11747:
11403:
5527:(Evaluated using the preceding series for
2314:
2146:
917:Kerala school of astronomy and mathematics
897:Kerala school of astronomy and mathematics
370:
356:
22229:
22044:
21933:Rabinowitz, Stanley; Wagon, Stan (1995).
21905:On-Line Encyclopedia of Integer Sequences
21880:On-Line Encyclopedia of Integer Sequences
21797:
21795:
21793:
21791:
21789:
21768:Borwein, Jonathan; Bailey, David (2008).
21729:
21668:
21666:
21632:"The Ratio of Proton and Electron Masses"
21521:
21486:
21307:
21164:
21078:"Continued Fraction Approximations to Pi"
20812:
20783:"y-cruncher: A Multi-Threaded Pi Program"
20640:"Announcement at the Kanada lab web site"
20440:
20173:
19887:
19876:Missouri Journal of Mathematical Sciences
19595:
19543:
19506:
19140:
17690:project computed 64 bits around the
15589:
15460:
14910:{\displaystyle a_{k}={\sqrt {2+a_{k-1}}}}
13583:
12650:
10332:Like the one above, a consequence of the
10228:) = 1, and d = 163 is the largest one in
6973:{\displaystyle {\frac {22}{7}}=3.143^{+}}
4958:
4918:
3130:
2942:On 14 August 2021, a team (DAViS) at the
2936:machines. This took 121 days to complete.
1741:
480:Some Egyptologists have claimed that the
22238:
21935:"A Spigot Algorithm for the Digits of Ο"
21527:
21366:
21351:
21328:
21289:
20758:
20749:
20740:
20541:"William Shanks (1812β1882) β Biography"
20412:
20326:
19849:"Ghiyath al-Din Jamshid Mas'ud al-Kashi"
19780:
19443:
19431:
19332:
19048:community. PiFast 4.4 is available from
13843:, the formula can be simplified to get:
13410:Note that Madhava's correction term is
12346:
11546:{\displaystyle d={\sqrt {x^{2}+y^{2}}}.}
11407:
7681:{\displaystyle {\sqrt{306}}=3.14155^{+}}
2619:to over 1 billion decimal places on the
2245:
1404:
800:. Zu Chongzhi is known to have computed
45:
22038:
21580:
21561:
21384:The Mathematical Papers of Isaac Newton
21050:
20767:
20255:
19869:
19339: β 3 from the early times of
19265:
19263:
14721:This series is the basis for a decimal
9483:= 151931373056001/151931373056000 β 1).
7178:{\displaystyle 1+e-\gamma =3.1410^{+},}
2594:United States Naval Research Laboratory
423:), and 126 digits by the 19th century (
22352:
21786:
21663:
21544:
21374:
21217:. recoveredscience.com. Archived from
21024:
20998:
20496:
20471:
19737:. Bangalore: Jnana Deep Publications.
19714:, School of Mathematics and Statistics
19690:Geometry: Seeing, Doing, Understanding
19687:
19581:
19488:
19397:
19356:
19223:
19208:
19182:
12508:
12351:This circle as it would be drawn on a
7269:{\displaystyle {\sqrt{31}}=3.1413^{+}}
5875:Ramanujan's work is the basis for the
3632:
430:The record of manual approximation of
22163:. University of Tokyo. Archived from
22014:
21744:
21675:Archive for History of Exact Sciences
21672:
21476:
21412:
21390:
21203:, New York: Dover Publications, 1993.
21053:"Best Rational Approximations for Pi"
20806:
20743:"Round 2... 10 Trillion Digits of Pi"
20714:
20262:Archive for History of Exact Sciences
20231:
19953:
19947:
19812:
19302:On Pyramid Dimensions and Proportions
19298:
19269:
19153:
17446:Extremely long decimal expansions of
12367:, so the approximate area is 81, and
11951:is 5, then the cells considered are:
11053:{\displaystyle \tau ={\sqrt {-3502}}}
5891:Extremely long decimal expansions of
3514:. Using these last values he obtains
3149:equals three", based on a passage in
2295:expressed to just 62 decimal places.
541:At about the same time, the Egyptian
22273:
21629:
21562:Anthony, Sebastian (15 March 2012).
21479:The Borwein brothers, Pi and the AGM
21401:Institutiones Calculi Differentialis
21316:
21301:
21277:
21248:from the original on 30 October 2007
21233:
21142:"On the rabbinical approximation of
20829:"Google Cloud Topples the Pi Record"
20302:
19260:
17477:published a paper (Bailey, 1997) on
17426:
6326:{\displaystyle f(y)=(1-y^{4})^{1/4}}
5886:
4821:/ Euler Convergence Transformation:
3137:Molten Sea Β§ Approximation of Ο
2961:over 158 days using Alexander Yee's
30:for a tabular summary. See also the
16:Varying methods used to calculate pi
22123:
21609:A nested radical approximation for
21342:
20973:Yee, Alexander J. (14 March 2024).
20972:
20780:
19872:"al-RisΔla al-muhΔ«tΔ«yya: A Summary"
19479:Indian Book Company (1975). p. 133.
18848:{\displaystyle O(M(n)(\log n)^{2})}
12381:= 3.24. Results for some values of
11485:gives the distance from any point (
11180:can be used to generate successive
3868:Formulae of this kind are known as
3854:) = {239, 13} is a solution to the
3026:; 'approximate ratio') and
13:
21623:
21085:Illinois Department of Mathematics
20938:
20174:
19834:Dictionary of Scientific Biography
19416:Bruins, E. M.; Rutten, M. (1961).
18969:Programs designed for calculating
18964:
18941:are also included in many general
18917:
18526:
18352:
18131:
18069:
17757:
17521:
17443:in England for a number of years.
17157:
17018:
16799:
16561:
16540:
16071:. However, these two formulae for
15805:
15192:
15002:
14410:
14365:
14269:
14193:
13626:
11701:
11388:{\displaystyle {\frac {355}{113}}}
11027:, and where the argument involves
5736:
5596:
5256:
5194:
4892:
4619:
4546:
3621:The last major attempt to compute
3169:and a circumference of 30 cubits.
3098:The bill was nearly passed by the
3044:Non-mathematical "definitions" of
2889:to calculate 5 trillion digits of
2672:
2391:
2325:In 1910, the Indian mathematician
2298:The English amateur mathematician
2276:) to a precision of less than one
1913:
1849:
1427:is the approximation after taking
1235:
1162:
931:. One of them is now known as the
553:text) implies an approximation of
14:
22391:
21939:The American Mathematical Monthly
21920:"Fractional Approximations of Pi"
21212:
20683:McCormick Grad Sets New Pi Record
20413:Ferguson, D. F. (16 March 1946).
20234:"William Jones: The First Use of
19902:
19559:Lazarus Mudehwe (February 1997).
19255:around and 280 cubits in height).
14947:{\displaystyle a_{1}={\sqrt {2}}}
11955:
11481:will fall inside the circle. The
3602:Advances in the approximation of
3269:Polygon approximation to a circle
3258:Development of efficient formulae
3141:It is sometimes claimed that the
2928:, computed 31.4 (approximately 10
2857:
972:{\displaystyle \pi =4\arctan(1):}
740:{\displaystyle 3+8/60+30/60^{2},}
574:Astronomical calculations in the
465:The best known approximations to
22251:. New York: St. Martin's Press.
20244:. McGrawβHill. pp. 346β347.
19489:Jadhav, Dipak (1 January 2018).
17450:are typically computed with the
13801:, which leads to formulae where
6382:The first one million digits of
2842:and a team of 9 others used the
2555:have been done with the help of
1442:correct to 11 decimal places as
460:
22152:
22139:
22117:
22095:
22065:
22053:
22032:
22007:
21973:
21926:
21912:
21887:
21862:
21837:
21820:
21761:
21738:
21701:
21630:Lenz, Friedrich (15 May 1951).
21602:
21574:
21555:
21334:
21322:
21295:
21283:
21271:
21260:
21206:
21188:
21133:
21106:
21070:
21044:
21025:Allain, Rhett (18 March 2011).
21018:
20999:Ranous, Jordan (28 June 2024).
20992:
20966:
20910:
20889:
20867:
20846:
20821:
20800:
20734:
20708:
20694:
20676:
20662:
20632:
20561:
20490:
20482:. p. viii – via the
20465:
20383:
20296:
20249:
19921:
19896:
19863:
19840:
19825:
19806:
19751:
19726:
19700:
19650:
19575:
19552:
19531:
19482:
19469:
19447:(2007). Katz, Victor J. (ed.).
19437:
19424:
19409:
19183:Ranous, Jordan (28 June 2024).
19154:Hayes, Brian (September 2014).
18774:{\displaystyle O(n\log(n)^{3})}
17439:. He worked with mathematician
14835:
14834:
13578:
13363:
10543:accurate to 161 decimal places:
6661:F. C. M. StΓΈrmer
5911:, converges extremely quickly:
5903:. The latter, found in 1985 by
3394:
3393:
3080:
1761:The German-Dutch mathematician
847:, in his astronomical treatise
22312:Jackson, K; Stamp, J. (2002).
21848:. Springer. pp. 596β622.
21846:Pi: A Source Book, 3rd Edition
21806:. Springer. pp. 241β257.
21804:Pi: A Source Book, 3rd Edition
21722:10.1080/00029890.1989.11972263
21267:Math Forum β Ask Dr. Math
21127:10.1080/0025570X.1977.11976632
19960:Synopsis Palmariorum Matheseos
19391:
19368:American Philosophical Society
19350:
19321:
19292:
19217:
19202:
19176:
19003:world record numbers of digits
18947:arbitrary-precision arithmetic
18842:
18833:
18820:
18817:
18811:
18805:
18768:
18759:
18752:
18740:
18712:{\displaystyle O(M(n)\log(n))}
18706:
18703:
18697:
18688:
18682:
18676:
18613:
18603:
18594:
18584:
18578:
18569:
18564:
18549:
18543:
18534:
18405:
18395:
18390:
18375:
18369:
18360:
18179:
18164:
18100:
18085:
17775:
17766:
17175:
17165:
17063:
17054:
16751:BaileyβBorweinβPlouffe formula
16608:
16593:
15652:
15646:
15586:
15580:
15558:
15552:
15523:
15517:
15457:
15451:
15429:
15423:
15396:
15390:
15355:
15349:
15167:
15161:
15111:
15091:
15059:
15044:
15033:
15023:
14980:
14974:
14445:
14430:
14335:
14320:
14236:
14221:
13649:
13639:
13394:
13364:
13205:
13195:
13149:
13139:
12523:generalized continued fraction
12385:are shown in the table below:
11698:
10994:
10975:
10946:
10927:
10910:
10884:
10855:
10836:
10784:
10755:
10752:
10723:
10714:
10684:
10675:
10645:
10582:
10572:
10486:
10471:
10467:
10451:
10448:
10428:
10425:
10405:
10402:
10386:
10383:
10367:
10345:accurate to 52 decimal places:
10340:the largest in absolute value.
10307:
10292:
10272:
10259:
10237:accurate to 52 decimal places:
10202:Derived from the closeness of
10134:
10115:
10093:accurate to 30 decimal places:
9998:
9976:
9929:accurate to 25 decimal places:
9353:accurate to 12 decimal places:
8887:
8873:
6306:
6286:
6280:
6274:
6248:
6199:
6152:
6126:
6085:
6082:
6069:
6057:
6049:
6046:
6033:
6021:
5823:
5813:
5807:
5798:
5793:
5778:
5772:
5763:
5754:
5744:
5649:
5639:
5634:
5619:
5613:
5604:
5316:
5301:
5225:
5210:
5139:
5119:
5065:
5045:
4979:
4959:
4949:
4934:
4909:
4900:
4644:
4627:
4564:
4554:
3828:
3816:
3787:
3775:
3763:
3750:
3589:Claudius Ptolemy of Alexandria
3172:The issue is discussed in the
3036:
3022:
2759:
2749:
2743:
2734:
2729:
2714:
2708:
2699:
2690:
2680:
2444:
2434:
2429:
2414:
2408:
2399:
1546:
1536:
1260:
1243:
1180:
1170:
1108:
1090:
963:
957:
906:
644:proved the sharp inequalities
1:
22231:10.1090/S0025-5718-97-00856-9
22183:
22161:"Kanada Laboratory home page"
21051:John D., Cook (22 May 2018).
20329:"Why 140 Digits of Pi Matter"
19870:Azarian, Mohammad K. (2010).
17704:further improved on BBP with
13828:
12363:the circle, and 69 cells are
11943:to find the approximation of
923:for arctangent, and then two
843:(6th century), mathematician
444:Chronology of computation of
25:chronology of computation of
21772:. A.K. Peters. p. 135.
20761:"12.1 Trillion Digits of Pi"
20242:A Source Book in Mathematics
20232:Smith, David Eugene (1929).
20162:
20097:
20032:
19963:. London: J. Wale. pp.
19692:(Third ed.). New York:
19617:. Springer. pp. 20β35.
19597:10.1016/0315-0860(86)90055-8
19477:A profile of Indian culture.
19418:Textes mathΓ©matiques de Suse
11459:{\displaystyle A=\pi r^{2}.}
11182:best rational approximations
6681:Miscellaneous approximations
3264:List of formulae involving Ο
3157:giving measurements for the
1433:(click for detail)
901:incommensurable (irrational)
873:to four decimal places: Ο β
806:two other approximations of
7:
22380:Real transcendental numbers
22103:"The world of Pi β Bellard"
21828:College Mathematics Journal
21586:New Mathematical Diversions
19836:. Vol. 7. p. 256.
19733:S. Balachandra Rao (1998).
19666:
19658:
19106:
18873:
17698:(which turns out to be 0).
17431:Many other expressions for
11412:Numerical approximation of
3165:as having a diameter of 10
2240:to calculate 100 digits of
886:
10:
22396:
22293:Joseph, George G. (2000).
22210:Mathematics of Computation
22105:. Pi314.net. 13 April 2013
21895:Sloane, N. J. A.
21870:Sloane, N. J. A.
20781:Yee, Alexander J. (2018).
20606:Mathematics of Computation
19792:"Madhava of Sangamagramma"
19764:Cambridge University Press
19688:Jacobs, Harold R. (2003).
19453:Princeton University Press
19276:Cambridge University Press
18937:Functions for calculating
18934:to any desired precision.
16094:
13839:Knowing that 4 arctan 1 =
13832:
13573:The Wolfram Functions Site
9287:accurate to twelve digits:
9221:accurate to eleven digits:
3725:expansion of the function
3636:
3629:using Snell's refinement.
3261:
3225:states (ca. 1168 CE) that
3134:
3125:postponing it indefinitely
3084:
3040:; 'close ratio').
2568:mechanical desk calculator
2318:
2246:Β§ Machin-like formula
547:Second Intermediate Period
543:Rhind Mathematical Papyrus
18:
21994:10.1017/S0025557200178404
21545:Kanada, Yasumasa (1996).
21497:10.1007/978-3-030-36568-4
21451:10.1017/S0025557200178404
20715:Glenn (19 October 2011).
20688:28 September 2011 at the
20327:Sandifer, Edward (2006).
19857:University of St. Andrews
19800:University of St. Andrews
19123:Madhava's correction term
19113:Diophantine approximation
18951:Class Library for Numbers
18895:Software for calculating
18878:
18654:Time complexity or Speed
16957:10 math to extract a base
8411:accurate to eight digits:
8066:accurate to seven digits:
6986:accurate to three digits:
6934:accurate to three digits:
3031:
3017:
2950:to 62.8 (approximately 20
747:which is accurate to two
484:used an approximation of
162:Madhava's correction term
21982:The Mathematical Gazette
21657:10.1103/PhysRev.82.554.2
21528:Hemphill, Scott (1993).
21439:The Mathematical Gazette
20702:"Pi β 5 Trillion Digits"
20600:(1962). "Calculation of
20546:University of St Andrews
20511:Royal Society Publishing
20497:Shanks, William (1873).
20472:Shanks, William (1853).
20342:]. Ljubljana: DMFA.
20217:; and by means thereof,
19889:10.35834/mjms/1312233136
19712:University of St Andrews
19694:W.H. Freeman and Company
19508:10.25078/ijhsrs.v2i1.511
19299:Legon, J. A. R. (1991).
19133:
18924:computer algebra systems
18788:of the arctan series in
18660:GaussβLegendre algorithm
17452:GaussβLegendre algorithm
16745:Digit extraction methods
14725:by Rabinowitz and Wagon.
13771:is the power series for
12494:For related results see
11469:If a circle with radius
8819:accurate to nine digits:
7585:accurate to five digits:
7207:natural logarithmic base
7131:accurate to four digits:
5897:GaussβLegendre algorithm
4336:Other classical formulae
3161:located in front of the
3100:Indiana General Assembly
3066:") and a passage in the
2984:Practical approximations
761:in 263 CE computed
640:In the 3rd century BCE,
453:with 202 trillion (2.02Γ
341:Other topics related to
21380:Whiteside, Derek Thomas
21368:How Euler Did Even More
21057:John D. Cook Consulting
20670:"Pi Computation Record"
19644:15 January 2017 at the
19329:#Imputed biblical value
19211:Wisdom of the Egyptians
19209:Petrie, W.M.F. (1940).
19156:"Pencil, Paper, and Pi"
17460:SalaminβBrent algorithm
11404:Summing a circle's area
11077:accurate to 256 digits:
9037:accurate to ten digits:
8977:accurate to ten digits:
7694:accurate to six digits:
7222:{\displaystyle \gamma }
6909:3.141592653589793237...
6361:{\displaystyle 1/a_{k}}
3591:to obtain the value of
3275:Measurement of a Circle
2315:20th and 21st centuries
1068:The other was based on
913:Madhava of Sangamagrama
675:In the 2nd century CE,
413:Madhava of Sangamagrama
22365:History of mathematics
21166:10.1006/hmat.1997.2185
20604:to 100,000 decimals".
20519:10.1098/rspl.1872.0066
20303:Vega, GΓ©orge (1795) .
20190:
19398:Bruins, E. M. (1950).
18932:mathematical constants
18849:
18775:
18713:
18632:
18530:
18434:
18356:
18282:
18135:
18073:
18017:
17761:
17653:
17525:
17417:
17161:
17104:2 math) for computing
17076:
17022:
16936:
16803:
16753:(BBP) for calculating
16732:
16544:
16150:
16085:
16057:
16027:
15928:
15809:
15760:
15737:
15717:
15697:
15674:
15315:
15196:
15136:
15006:
14948:
14911:
14852:
14745:
14713:
14597:
14369:
14273:
14197:
13815:
13795:
13762:
13630:
13590:GregoryβLeibniz series
13584:GregoryβLeibniz series
13551:The well-known values
13541:
13401:
13135:
13094:MadhavaβLeibniz series
13083:
12671:
12356:
11921:are integers between β
11897:. Consider all cells (
11884:
11771:
11746:
11654:
11547:
11460:
11421:
11389:
11358:
11159:
11113:
11054:
11015:Based on one found by
11005:
10791:
10616:
10519:
10322:
10192:
10068:
9916:
9871:1029347477390786609545
9860:2286635172367940241408
9843:accurate to 24 digits:
9830:
9763:accurate to 19 digits:
9742:
9627:accurate to 18 digits:
9615:
9555:
9488:accurate to 16 digits:
9442:
9341:
9275:
9198:
9126:
9025:
8950:
8804:
8676:
8629:
8546:
8473:
8399:
8231:
8175:
8112:
8054:
7870:
7785:
7682:
7639:
7573:
7502:
7433:
7378:
7324:
7270:
7223:
7199:
7179:
7119:
7033:
6974:
6915:
6786:
6653:
6530:
6362:
6327:
6255:
5986:
5866:
5740:
5678:
5600:
5519:
5425:
5260:
5198:
4896:
4809:
4623:
4550:
4495:
4320:
4199:
4075:
3976:
3838:
3712:
3566:
3445:
3187:explained this in his
3131:Imputed biblical value
2865:T2K Open Supercomputer
2805:
2676:
2535:
2473:
2395:
2234:
2154:
1917:
1853:
1742:16th to 19th centuries
1725:
1693:
1625:
1435:
1396:
1239:
1166:
1118:
1059:
973:
933:MadhavaβLeibniz series
867:
741:
668:, by means of regular
638:
506:Babylonian mathematics
398:history of mathematics
66:mathematical constant
51:
22274:Eves, Howard (1992).
21477:Trueb, Peter (2020).
21352:Sandifer, Ed (2009).
21343:Roy, Ranjan (2021) .
21005:www.storagereview.com
20951:Google Cloud Platform
20238:for the Circle Ratio"
20191:
19708:"Aryabhata the Elder"
19249:Great Pyramid of Giza
19189:www.storagereview.com
18860:Leibniz formula for Ο
18850:
18776:
18714:
18633:
18510:
18435:
18336:
18283:
18115:
18053:
18018:
17741:
17684:programming languages
17654:
17505:
17418:
17141:
17077:
17002:
16937:
16783:
16733:
16524:
16151:
16086:
16058:
16056:{\displaystyle F_{n}}
16028:
15929:
15789:
15761:
15738:
15718:
15698:
15675:
15316:
15176:
15137:
14986:
14949:
14912:
14853:
14746:
14714:
14598:
14349:
14253:
14177:
13833:Further information:
13816:
13796:
13763:
13610:
13542:
13402:
13115:
13092:The remainder of the
13084:
12672:
12350:
11885:
11748:
11723:
11655:
11548:
11461:
11411:
11390:
11359:
11160:
11114:
11055:
11025:Dedekind eta function
11006:
10792:
10617:
10520:
10323:
10193:
10086:= 2/(5 β 1)
10069:
9917:
9831:
9743:
9616:
9556:
9443:
9342:
9276:
9199:
9127:
9026:
8951:
8805:
8677:
8630:
8547:
8474:
8400:
8232:
8176:
8113:
8055:
7871:
7786:
7683:
7640:
7574:
7503:
7434:
7379:
7325:
7271:
7224:
7200:
7180:
7120:
7034:
6975:
6916:
6787:
6654:
6531:
6369:converges quartically
6363:
6328:
6256:
5987:
5867:
5720:
5679:
5580:
5520:
5426:
5240:
5178:
4876:
4810:
4603:
4530:
4496:
4321:
4200:
4076:
3977:
3839:
3713:
3567:
3446:
2917:(22,459,157,718,361 (
2838:. In November 2002,
2806:
2656:
2546:RamanujanβSato series
2536:
2474:
2375:
2235:
2155:
1897:
1833:
1726:
1694:
1626:
1408:
1397:
1219:
1146:
1119:
1060:
974:
853:
756:Chinese mathematician
742:
625:"verses: 6.12.40β45,
606:
508:usually approximated
471:before the Common Era
387:mathematical constant
115:Use in other formulae
49:
21428:: 133β149, 167β168.
21195:Wilbur Richard Knorr
21153:Historia Mathematica
21115:Mathematics Magazine
20480:Macmillan Publishers
19998:
19584:Historia Mathematica
19475:Chaitanya, Krishna.
19358:Romano, David Gilman
19347:in the 2nd century."
19170:10.1511/2014.110.342
18799:
18734:
18724:Chudnovsky algorithm
18670:
18475:
18466:Chudnovsky algorithm
18301:
18037:
17715:
17496:
17441:Godfrey Harold Hardy
17115:
16987:
16949:th decimal digit of
16774:
16166:
16106:
16084:{\displaystyle \pi }
16075:
16040:
15944:
15773:
15759:{\displaystyle \pi }
15750:
15727:
15723:improves as integer
15716:{\displaystyle \pi }
15707:
15696:{\displaystyle \pi }
15687:
15331:
15152:
14965:
14921:
14865:
14758:
14744:{\displaystyle \pi }
14735:
14731:Another formula for
14613:
13850:
13823:Machin-like formulae
13814:{\displaystyle \pi }
13805:
13785:
13598:
13417:
13103:
12682:
12532:
12353:Cartesian coordinate
11681:
11611:
11504:
11431:
11420:as points are added.
11372:
11198:
11124:
11087:
11031:
10807:
10636:
10553:
10355:
10247:
10103:
9939:
9853:
9773:
9637:
9567:
9498:
9363:
9297:
9231:
9148:
9047:
8987:
8829:
8687:
8640:
8557:
8484:
8421:
8243:
8186:
8123:
8076:
7881:
7798:
7704:
7650:
7595:
7528:
7514:an approximation by
7455:
7389:
7335:
7281:
7238:
7213:
7189:
7141:
7076:
6996:
6944:
6803:
6704:
6548:
6425:
6337:
6268:
5999:
5918:
5877:Chudnovsky algorithm
5701:
5545:
5463:
4828:
4514:
4361:
4215:
4094:
3992:
3896:
3871:Machin-like formulae
3747:
3654:
3521:
3317:
3191:(the earliest known
2822:and his team at the
2816:Chudnovsky algorithm
2637:
2497:
2340:
2185:
1817:
1709:
1671:
1465:
1130:
1072:
985:
939:
693:
604:verses: 6.12.40β45.
59:a series of articles
22222:1997MaCom..66..903B
22059:Bellard's Website:
21751:Amer. Math. Monthly
21710:Amer. Math. Monthly
21649:1951PhRv...82..554L
21617:6 July 2011 at the
20644:Super-computing.org
20433:1946Natur.157..342F
20375:on 28 August 2006.
19386:was 3 1/8 or 3.125.
18443:Srinivasa Ramanujan
17456:Borwein's algorithm
17437:Srinivasa Ramanujan
15292:
15263:
14400:
14375:
14317:
14279:
14218:
14203:
14118:
14082:
14045:
14015:
13984:
13960:
13935:
13917:
13898:
13886:
13775:(x) specialized to
13664:
13636:
13328:
13309:
13292:
13273:
13256:
13237:
13220:
13189:
13042:
13023:
13009:
12990:
12976:
12957:
12943:
12924:
12910:
12891:
12840:
12821:
12807:
12788:
12774:
12755:
12741:
12722:
12708:
12696:
12638:
12619:
12605:
12586:
12572:
12553:
12513:Besides its simple
12509:Continued fractions
11483:Pythagorean theorem
11060:. The discriminant
8966:Goddess of Namagiri
6398:are available from
6247:
5901:Borwein's algorithm
5298:
5266:
3639:Machin-like formula
3633:Machin-like formula
3273:Archimedes, in his
3178:Rabbinic literature
3163:Temple in Jerusalem
3093:squaring the circle
3006:of about 4Β·10) and
2824:University of Tokyo
2613:Chudnovsky brothers
2327:Srinivasa Ramanujan
1786:Willebrord Snellius
893:Nilakantha Somayaji
577:Shatapatha Brahmana
475:Chinese mathematics
406:Chinese mathematics
323:Squaring the circle
258:Chudnovsky brothers
248:Srinivasa Ramanujan
22124:Bellard, Fabrice.
22016:Weisstein, Eric W.
21908:. OEIS Foundation.
21883:. OEIS Foundation.
21687:10.1007/BF00348349
21221:on 14 October 2007
21094:on 23 January 2021
20274:10.1007/BF00384331
20186:
20159:
20137:
20115:
20094:
20072:
20050:
20029:
20014:
19659:gaαΉitapΔda 10
19571:on 8 January 2013.
19341:rabbinical Judaism
19161:American Scientist
18912:personal computers
18845:
18771:
18709:
18628:
18462:Gregory Chudnovsky
18430:
18278:
18013:
17649:
17413:
17072:
16932:
16728:
16726:
16146:
16081:
16053:
16023:
15975:
15924:
15756:
15733:
15713:
15693:
15670:
15668:
15311:
15278:
15249:
15132:
14944:
14907:
14848:
14741:
14709:
14593:
14591:
14449:
14395:
14342:
14312:
14246:
14213:
14149:
14113:
14070:
14040:
14003:
13979:
13948:
13930:
13905:
13893:
13811:
13791:
13758:
13680:
13659:
13537:
13397:
13359:
13354:
13349:
13344:
13323:
13287:
13251:
13215:
13079:
13075:
13070:
13065:
13060:
13055:
13037:
13004:
12971:
12938:
12905:
12873:
12868:
12863:
12858:
12853:
12835:
12802:
12769:
12736:
12703:
12667:
12662:
12657:
12652:
12633:
12600:
12567:
12515:continued fraction
12357:
11947:. For example, if
11880:
11875:
11705:
11650:
11543:
11456:
11422:
11385:
11354:
11176:representation of
11174:continued fraction
11155:
11109:
11050:
11001:
10999:
10973:
10882:
10834:
10787:
10612:
10515:
10318:
10204:Ramanujan constant
10188:
10064:
9912:
9826:
9738:
9611:
9551:
9438:
9337:
9271:
9194:
9122:
9021:
8964:, who claimed the
8946:
8800:
8672:
8625:
8542:
8469:
8395:
8227:
8171:
8108:
8050:
7866:
7781:
7678:
7635:
7569:
7498:
7429:
7374:
7320:
7266:
7219:
7195:
7175:
7115:
7029:
6970:
6911:
6782:
6649:
6526:
6358:
6323:
6251:
6227:
5982:
5862:
5692:Gregory Chudnovsky
5674:
5515:
5421:
5419:
5323:
5293:
4805:
4491:
4489:
4485:3.141590463236763.
4316:
4208:K. Takano (1982).
4195:
4071:
3972:
3834:
3721:together with the
3708:
3562:
3441:
3070:that implies that
2954:) trillion digits.
2874:In December 2009,
2832:HITACHI SR8000/MPP
2801:
2531:
2469:
2262:William Rutherford
2230:
2198:
2150:
1790:Christiaan Huygens
1763:Ludolph van Ceulen
1738:with 3 Γ 2 sides.
1721:
1719:3.1415926535897932
1689:
1684:6.2831853071795864
1650:Persian astronomer
1621:
1436:
1392:
1114:
1055:
969:
737:
421:Ludolph van Ceulen
213:Ludolph van Ceulen
52:
22342:978-1-4757-4217-6
22333:Pi: a source book
22304:978-0-14-027778-4
22285:978-0-03-029558-4
22258:978-0-88029-418-8
22195:Borwein, Peter B.
22167:on 24 August 2011
21855:978-0-387-20571-7
21813:978-0-387-20571-7
21779:978-1-56881-442-1
21595:978-0-88385-517-1
21548:One Divided by Pi
21506:978-3-030-36567-7
21242:"A history of Pi"
21213:Aleff, H. Peter.
20598:Wrench, J. W. Jr.
20349:978-961-6137-98-0
20313:figures decimals"
20165:
20158:
20136:
20114:
20100:
20093:
20071:
20049:
20035:
20028:
20013:
19773:978-0-521-78988-2
19760:Special Functions
19744:978-81-7371-205-0
19614:Pi: A Source Book
19561:"The story of pi"
19462:978-0-691-11485-9
19445:Imhausen, Annette
19285:978-0-521-69053-9
19242:978-0-8021-3935-1
19100:Super PI 1.9 page
19019:Microsoft Windows
18930:and other common
18871:
18870:
18626:
18508:
18505:
18486:
18428:
18334:
18328:
18312:
18247:
18226:
18205:
18186:
18110:
18048:
18006:
17982:
17951:
17920:
17889:
17858:
17834:
17798:
17739:
17639:
17615:
17591:
17567:
17541:
17427:Efficient methods
17406:
17382:
17351:
17320:
17289:
17258:
17234:
17198:
17139:
17070:
16920:
16900:
16876:
16852:
16828:
16716:
16703:
16690:
16677:
16664:
16651:
16638:
16612:
16573:
16512:
16484:
16456:
16428:
16381:
16321:
16273:
16239:
16214:
16144:
16127:
16018:
16007:
15960:
15955:
15916:
15897:
15878:
15859:
15840:
15784:
15736:{\displaystyle k}
15305:
15217:
15127:
15066:
14942:
14905:
14829:
14818:
14782:
14678:
14657:
14636:
14581:
14568:
14555:
14542:
14529:
14516:
14503:
14490:
14477:
14451:
14422:
14399:
14374:
14344:
14316:
14278:
14248:
14217:
14202:
14151:
14117:
14081:
14072:
14044:
14014:
14005:
13983:
13959:
13950:
13934:
13916:
13907:
13897:
13885:
13794:{\displaystyle x}
13742:
13729:
13716:
13703:
13682:
13663:
13635:
13535:
13483:
13480:
13477:
13361:
13356:
13351:
13346:
13327:
13308:
13291:
13272:
13255:
13236:
13219:
13188:
13179:
13077:
13072:
13067:
13062:
13057:
13041:
13022:
13008:
12989:
12975:
12956:
12942:
12923:
12909:
12890:
12875:
12870:
12865:
12860:
12855:
12839:
12820:
12806:
12787:
12773:
12754:
12740:
12721:
12707:
12695:
12664:
12659:
12654:
12637:
12618:
12604:
12585:
12571:
12552:
12492:
12491:
12397:approximation of
12365:completely inside
12343:
12342:
11862:
11835:
11815:
11788:
11721:
11690:
11639:
11538:
11497:) to the center:
11383:
11352:
11339:
11326:
11313:
11300:
11287:
11274:
11261:
11248:
11235:
11222:
11209:
11153:
11147:
11098:
11048:
10992:
10972:
10944:
10908:
10895:
10881:
10853:
10833:
10782:
10750:
10711:
10672:
10610:
10609:
10513:
10510:
10503:
10496:
10459:
10446:
10436:
10423:
10413:
10400:
10316:
10315:
10289:
10177:
10171:
10165:
10159:
10153:
10143:
10142:
10053:
10047:
10041:
10035:
10029:
10008:
9989:
9950:
9901:
9895:
9889:
9883:
9873:
9867:
9815:
9809:
9803:
9793:
9787:
9727:
9721:
9715:
9691:
9678:
9672:
9656:
9652:
9600:
9594:
9588:
9578:
9540:
9534:
9528:
9518:
9512:
9427:
9421:
9397:
9396:
9382:
9378:
9326:
9320:
9310:
9260:
9254:
9244:
9183:
9173:
9167:
9111:
9101:
9098:
9080:
9058:
9010:
9000:
8935:
8925:
8919:
8905:
8897:
8884:
8789:
8779:
8770:
8758:
8728:
8722:
8706:
8702:
8661:
8651:
8614:
8604:
8598:
8586:
8574:
8531:
8521:
8518:
8506:
8458:
8448:
8442:
8432:
8384:
8369:
8366:
8363:
8287:
8216:
8206:
8203:
8160:
8150:
8144:
8134:
8097:
8087:
8039:
8029:
8026:
8023:
8020:
7945:
7932:
7919:
7906:
7855:
7845:
7839:
7829:
7816:
7806:
7770:
7749:
7738:
7730:
7663:
7620:
7554:
7553:
7539:
7483:
7481:
7479:
7418:
7408:
7404:
7363:
7353:
7343:
7309:
7299:
7289:
7251:
7198:{\displaystyle e}
7094:
7084:
7046:conjectured that
7014:
7004:
6955:
6903:
6883:
6863:
6843:
6823:
6771:
6761:
6741:
6721:
6647:
6625:
6603:
6581:
6559:
6538:Kikuo Takano
6524:
6502:
6480:
6458:
6436:
6400:Project Gutenberg
6096:
6090:
5980:
5952:
5939:
5887:Modern algorithms
5860:
5712:
5672:
5578:
5572:
5556:
5513:
5491:
5386:
5365:
5344:
5325:
5297:
5265:
5235:
5169:
5149:
5104:
5075:
5030:
5017:
4989:
4874:
4792:
4764:
4736:
4708:
4678:
4668:
4641:
4601:
4591:
4528:
4472:
4470:
4468:
4466:
4464:
4462:
4460:
4458:
4456:
4314:
4292:
4270:
4248:
4226:
4193:
4171:
4149:
4127:
4105:
4069:
4047:
4025:
4003:
3970:
3951:
3929:
3907:
3735:polar coordinates
3706:
3687:
3665:
3557:
3535:
3436:
3388:
3189:Mishnat ha-Middot
2796:
2648:
2523:
2520:
2467:
2373:
2367:
2351:
2248:below). In 1719,
2197:
2134:
2121:
2094:
2081:
2054:
2041:
2014:
2001:
1976:
1947:
1892:
1828:
1642: = 75.
1619:
1570:
1522:
1509:
1496:
1379:
1351:
1323:
1294:
1284:
1257:
1217:
1207:
1144:
1106:
1042:
1029:
1016:
915:, founder of the
482:ancient Egyptians
380:
379:
22387:
22346:
22327:
22308:
22289:
22270:
22248:
22235:
22233:
22216:(218): 903β913.
22207:
22191:Bailey, David H.
22177:
22176:
22174:
22172:
22156:
22150:
22143:
22137:
22136:
22134:
22132:
22121:
22115:
22114:
22112:
22110:
22099:
22093:
22092:
22090:
22088:
22083:on 10 April 2011
22079:. Archived from
22073:"David H Bailey"
22069:
22063:
22057:
22051:
22050:
22048:
22036:
22030:
22029:
22028:
22011:
22005:
22004:
21988:(516): 469β470,
21977:
21971:
21970:
21930:
21924:
21923:
21916:
21910:
21909:
21891:
21885:
21884:
21866:
21860:
21859:
21841:
21835:
21824:
21818:
21817:
21799:
21784:
21783:
21765:
21759:
21758:
21748:
21742:
21736:
21735:
21733:
21705:
21699:
21698:
21670:
21661:
21660:
21627:
21621:
21612:
21606:
21600:
21599:
21578:
21572:
21571:
21559:
21553:
21552:
21542:
21536:
21535:
21525:
21519:
21518:
21490:
21474:
21463:
21461:
21445:(516): 469β470,
21433:
21409:
21387:
21371:
21364:
21361:How Euler Did It
21358:
21348:
21338:
21332:
21331:, pp. 94β95
21326:
21320:
21314:
21305:
21299:
21293:
21287:
21281:
21275:
21269:
21264:
21258:
21257:
21255:
21253:
21237:
21231:
21230:
21228:
21226:
21210:
21204:
21192:
21186:
21185:
21183:
21181:
21168:
21150:
21145:
21137:
21131:
21130:
21110:
21104:
21103:
21101:
21099:
21093:
21082:
21074:
21068:
21067:
21065:
21063:
21048:
21042:
21041:
21039:
21037:
21022:
21016:
21015:
21013:
21011:
20996:
20990:
20989:
20987:
20985:
20970:
20964:
20963:
20961:
20959:
20942:
20936:
20935:
20933:
20931:
20926:. 16 August 2021
20914:
20908:
20907:
20905:
20903:
20893:
20887:
20886:
20884:
20882:
20871:
20865:
20864:
20862:
20860:
20850:
20844:
20843:
20841:
20839:
20825:
20819:
20818:
20816:
20804:
20798:
20797:
20795:
20793:
20778:
20765:
20764:
20756:
20747:
20746:
20738:
20732:
20731:
20729:
20727:
20712:
20706:
20705:
20698:
20692:
20680:
20674:
20673:
20666:
20660:
20659:
20657:
20655:
20650:on 12 March 2011
20646:. Archived from
20636:
20630:
20629:
20603:
20590:
20575:
20567:Ferguson 1946a,
20565:
20559:
20558:
20556:
20554:
20537:
20531:
20530:
20494:
20488:
20487:
20484:Internet Archive
20469:
20463:
20462:
20444:
20442:10.1038/157342c0
20410:
20401:
20400:
20387:
20381:
20379:
20374:
20368:. Archived from
20333:
20324:
20312:
20308:
20300:
20294:
20293:
20259:
20253:
20247:
20245:
20237:
20228:
20225:
20206:
20195:
20193:
20192:
20187:
20166:
20161:
20160:
20157:
20156:
20144:
20138:
20135:
20134:
20122:
20118:
20116:
20107:
20101:
20096:
20095:
20092:
20091:
20079:
20073:
20070:
20069:
20057:
20053:
20051:
20042:
20036:
20031:
20030:
20021:
20015:
20006:
20002:
19951:
19945:
19944:
19934:
19925:
19919:
19918:
19916:
19914:
19909:
19900:
19894:
19893:
19891:
19867:
19861:
19860:
19844:
19838:
19837:
19829:
19823:
19822:
19810:
19804:
19803:
19787:
19778:
19777:
19755:
19749:
19748:
19730:
19724:
19723:
19721:
19719:
19704:
19698:
19697:
19685:
19684:
19680:
19669:
19661:
19654:
19648:
19636:
19630:
19628:
19608:
19599:
19579:
19573:
19572:
19567:. Archived from
19556:
19550:
19549:
19547:
19535:
19529:
19528:
19510:
19486:
19480:
19473:
19467:
19466:
19441:
19435:
19428:
19422:
19421:
19413:
19407:
19406:
19404:
19395:
19389:
19388:
19385:
19354:
19348:
19338:
19325:
19319:
19318:
19316:
19314:
19296:
19290:
19289:
19267:
19258:
19257:
19225:Verner, Miroslav
19221:
19215:
19214:
19206:
19200:
19199:
19197:
19195:
19180:
19174:
19173:
19151:
19084:
19082:
19081:
19073:
19072:
19071:
19064:
19043:
19041:
19040:
19032:
19010:
18997:
18972:
18940:
18929:
18905:
18898:
18889:
18854:
18852:
18851:
18846:
18841:
18840:
18790:Machin's formula
18786:Binary splitting
18780:
18778:
18777:
18772:
18767:
18766:
18718:
18716:
18715:
18710:
18645:
18644:
18637:
18635:
18634:
18629:
18627:
18625:
18624:
18623:
18602:
18601:
18567:
18532:
18529:
18524:
18509:
18507:
18506:
18501:
18492:
18487:
18479:
18458:David Chudnovsky
18452:
18439:
18437:
18436:
18431:
18429:
18427:
18426:
18425:
18413:
18412:
18393:
18358:
18355:
18350:
18335:
18330:
18329:
18324:
18318:
18313:
18305:
18287:
18285:
18284:
18279:
18277:
18273:
18272:
18268:
18267:
18263:
18248:
18240:
18227:
18219:
18206:
18198:
18187:
18185:
18162:
18161:
18160:
18148:
18147:
18137:
18134:
18129:
18111:
18109:
18083:
18075:
18072:
18067:
18049:
18041:
18029:
18022:
18020:
18019:
18014:
18012:
18008:
18007:
18005:
17988:
17983:
17981:
17967:
17966:
17957:
17952:
17950:
17936:
17935:
17926:
17921:
17919:
17905:
17904:
17895:
17890:
17888:
17874:
17873:
17864:
17859:
17857:
17840:
17835:
17833:
17819:
17818:
17809:
17799:
17797:
17796:
17784:
17783:
17778:
17763:
17760:
17755:
17740:
17738:
17737:
17725:
17697:
17677:
17658:
17656:
17655:
17650:
17645:
17641:
17640:
17638:
17621:
17616:
17614:
17597:
17592:
17590:
17573:
17568:
17566:
17549:
17542:
17540:
17539:
17527:
17524:
17519:
17484:
17449:
17434:
17422:
17420:
17419:
17414:
17412:
17408:
17407:
17405:
17388:
17383:
17381:
17367:
17366:
17357:
17352:
17350:
17336:
17335:
17326:
17321:
17319:
17305:
17304:
17295:
17290:
17288:
17274:
17273:
17264:
17259:
17257:
17240:
17235:
17233:
17219:
17218:
17209:
17199:
17197:
17196:
17184:
17183:
17182:
17163:
17160:
17155:
17140:
17138:
17137:
17125:
17107:
17103:
17095:
17081:
17079:
17078:
17073:
17071:
17069:
17052:
17051:
17050:
17038:
17037:
17024:
17021:
17016:
16979:
16975:
16960:
16956:
16952:
16948:
16941:
16939:
16938:
16933:
16931:
16930:
16925:
16921:
16913:
16906:
16902:
16901:
16899:
16882:
16877:
16875:
16858:
16853:
16851:
16834:
16829:
16827:
16810:
16802:
16797:
16766:
16761:
16756:
16737:
16735:
16734:
16729:
16727:
16717:
16709:
16704:
16696:
16691:
16683:
16678:
16670:
16665:
16657:
16652:
16644:
16639:
16631:
16617:
16613:
16611:
16592:
16591:
16581:
16580:
16579:
16578:
16569:
16560:
16546:
16543:
16538:
16513:
16511:
16504:
16503:
16490:
16485:
16483:
16476:
16475:
16462:
16457:
16455:
16448:
16447:
16434:
16429:
16427:
16420:
16419:
16406:
16398:
16394:
16390:
16382:
16380:
16379:
16378:
16344:
16327:
16322:
16320:
16319:
16318:
16290:
16279:
16274:
16272:
16271:
16270:
16245:
16240:
16232:
16219:
16215:
16207:
16198:
16197:
16155:
16153:
16152:
16147:
16145:
16137:
16132:
16128:
16120:
16090:
16088:
16087:
16082:
16069:Fibonacci number
16062:
16060:
16059:
16054:
16052:
16051:
16032:
16030:
16029:
16024:
16019:
16017:
16016:
16006:
16005:
15984:
15983:
15974:
15956:
15948:
15933:
15931:
15930:
15925:
15917:
15909:
15898:
15890:
15879:
15871:
15860:
15852:
15841:
15839:
15838:
15817:
15808:
15803:
15785:
15777:
15765:
15763:
15762:
15757:
15742:
15740:
15739:
15734:
15722:
15720:
15719:
15714:
15702:
15700:
15699:
15694:
15679:
15677:
15676:
15671:
15669:
15659:
15645:
15644:
15623:
15619:
15618:
15617:
15608:
15579:
15578:
15551:
15550:
15540:
15530:
15516:
15515:
15494:
15490:
15489:
15488:
15479:
15450:
15449:
15422:
15421:
15411:
15389:
15388:
15378:
15368:
15348:
15347:
15337:
15320:
15318:
15317:
15312:
15307:
15306:
15304:
15303:
15291:
15286:
15274:
15262:
15257:
15247:
15246:
15235:
15234:
15233:
15228:
15220:
15218:
15216:
15199:
15195:
15190:
15141:
15139:
15138:
15133:
15128:
15126:
15125:
15124:
15109:
15108:
15089:
15088:
15070:
15067:
15065:
15042:
15041:
15040:
15022:
15021:
15008:
15005:
15000:
14953:
14951:
14950:
14945:
14943:
14938:
14933:
14932:
14916:
14914:
14913:
14908:
14906:
14904:
14903:
14882:
14877:
14876:
14857:
14855:
14854:
14849:
14830:
14828:
14827:
14817:
14816:
14795:
14794:
14783:
14781:
14780:
14762:
14750:
14748:
14747:
14742:
14723:spigot algorithm
14718:
14716:
14715:
14710:
14708:
14704:
14703:
14699:
14698:
14694:
14679:
14671:
14658:
14650:
14637:
14629:
14602:
14600:
14599:
14594:
14592:
14582:
14574:
14569:
14561:
14556:
14548:
14543:
14535:
14530:
14522:
14517:
14509:
14504:
14496:
14491:
14483:
14478:
14470:
14456:
14452:
14450:
14448:
14429:
14428:
14427:
14418:
14409:
14396:
14394:
14393:
14392:
14371:
14368:
14363:
14345:
14343:
14341:
14313:
14311:
14310:
14309:
14297:
14296:
14275:
14272:
14267:
14249:
14247:
14245:
14214:
14212:
14199:
14196:
14191:
14167:
14163:
14159:
14152:
14150:
14148:
14114:
14112:
14078:
14073:
14071:
14069:
14041:
14039:
14011:
14006:
14004:
14002:
13980:
13978:
13956:
13951:
13949:
13947:
13931:
13929:
13913:
13908:
13906:
13904:
13894:
13892:
13882:
13842:
13835:Double factorial
13820:
13818:
13817:
13812:
13800:
13798:
13797:
13792:
13780:
13767:
13765:
13764:
13759:
13757:
13753:
13743:
13735:
13730:
13722:
13717:
13709:
13704:
13696:
13683:
13681:
13679:
13660:
13658:
13657:
13656:
13632:
13629:
13624:
13570:
13569:
13565:
13560:
13559:
13555:
13546:
13544:
13543:
13538:
13536:
13534:
13524:
13523:
13510:
13503:
13502:
13492:
13484:
13482:
13481:
13479:
13478:
13476:
13468:
13467:
13458:
13446:
13445:
13436:
13421:
13406:
13404:
13403:
13398:
13362:
13360:
13358:
13357:
13355:
13353:
13352:
13350:
13348:
13347:
13345:
13343:
13324:
13322:
13321:
13320:
13305:
13288:
13286:
13285:
13284:
13269:
13252:
13250:
13249:
13248:
13233:
13216:
13214:
13213:
13212:
13185:
13180:
13178:
13164:
13163:
13162:
13137:
13134:
13129:
13088:
13086:
13085:
13080:
13078:
13076:
13074:
13073:
13071:
13069:
13068:
13066:
13064:
13063:
13061:
13059:
13058:
13056:
13054:
13038:
13036:
13035:
13034:
13019:
13005:
13003:
13002:
13001:
12986:
12972:
12970:
12969:
12968:
12953:
12939:
12937:
12936:
12935:
12920:
12906:
12904:
12903:
12902:
12887:
12876:
12874:
12872:
12871:
12869:
12867:
12866:
12864:
12862:
12861:
12859:
12857:
12856:
12854:
12852:
12836:
12834:
12833:
12832:
12817:
12803:
12801:
12800:
12799:
12784:
12770:
12768:
12767:
12766:
12751:
12737:
12735:
12734:
12733:
12718:
12704:
12702:
12692:
12676:
12674:
12673:
12668:
12666:
12665:
12663:
12661:
12660:
12658:
12656:
12655:
12653:
12651:
12634:
12632:
12631:
12630:
12615:
12601:
12599:
12598:
12597:
12582:
12568:
12566:
12565:
12564:
12549:
12520:
12504:
12400:
12388:
12387:
12384:
12380:
12379:
12375:
12370:
11956:
11950:
11946:
11942:
11938:
11933:
11928:
11924:
11920:
11914:
11909:) in which both
11908:
11902:
11896:
11889:
11887:
11886:
11881:
11879:
11878:
11863:
11861:
11860:
11848:
11847:
11838:
11836:
11833:
11816:
11814:
11813:
11801:
11800:
11791:
11789:
11786:
11770:
11765:
11745:
11740:
11722:
11720:
11719:
11707:
11704:
11670:
11666:
11659:
11657:
11656:
11651:
11640:
11638:
11637:
11625:
11624:
11615:
11603:
11597:
11591:
11587:
11579:
11573:
11567:
11561:
11552:
11550:
11549:
11544:
11539:
11537:
11536:
11524:
11523:
11514:
11496:
11490:
11479:
11473:
11465:
11463:
11462:
11457:
11452:
11451:
11419:
11415:
11398:
11394:
11392:
11391:
11386:
11384:
11376:
11363:
11361:
11360:
11355:
11353:
11345:
11340:
11332:
11327:
11319:
11314:
11306:
11301:
11293:
11288:
11280:
11275:
11267:
11262:
11254:
11249:
11241:
11236:
11228:
11223:
11215:
11210:
11202:
11187:
11179:
11164:
11162:
11161:
11156:
11154:
11149:
11148:
11143:
11137:
11118:
11116:
11115:
11110:
11099:
11091:
11059:
11057:
11056:
11051:
11049:
11041:
11010:
11008:
11007:
11002:
11000:
10993:
10988:
10974:
10965:
10945:
10940:
10909:
10904:
10896:
10891:
10883:
10874:
10854:
10849:
10835:
10826:
10796:
10794:
10793:
10788:
10783:
10775:
10774:
10765:
10751:
10743:
10742:
10733:
10722:
10721:
10712:
10704:
10703:
10694:
10683:
10682:
10673:
10665:
10664:
10655:
10621:
10619:
10618:
10613:
10611:
10605:
10604:
10603:
10602:
10590:
10589:
10571:
10570:
10557:
10537:
10524:
10522:
10521:
10516:
10514:
10512:
10511:
10506:
10504:
10499:
10497:
10492:
10489:
10479:
10478:
10460:
10455:
10447:
10442:
10437:
10432:
10424:
10419:
10414:
10409:
10401:
10396:
10382:
10381:
10359:
10327:
10325:
10324:
10319:
10317:
10311:
10310:
10300:
10299:
10290:
10285:
10271:
10270:
10251:
10197:
10195:
10194:
10189:
10187:
10186:
10175:
10169:
10163:
10157:
10151:
10144:
10138:
10137:
10127:
10126:
10107:
10087:
10073:
10071:
10070:
10065:
10063:
10062:
10051:
10045:
10039:
10033:
10027:
10020:
10016:
10009:
10007:
10006:
10005:
9990:
9988:
9980:
9974:
9973:
9964:
9951:
9943:
9921:
9919:
9918:
9913:
9911:
9910:
9899:
9893:
9887:
9881:
9874:
9869:
9868:
9863:
9857:
9835:
9833:
9832:
9827:
9825:
9824:
9813:
9807:
9801:
9794:
9789:
9788:
9783:
9777:
9757:
9747:
9745:
9744:
9739:
9737:
9736:
9725:
9719:
9713:
9706:
9705:
9697:
9693:
9692:
9684:
9679:
9674:
9673:
9668:
9662:
9657:
9648:
9647:
9620:
9618:
9617:
9612:
9610:
9609:
9598:
9592:
9586:
9579:
9571:
9560:
9558:
9557:
9552:
9550:
9549:
9538:
9532:
9526:
9519:
9514:
9513:
9508:
9502:
9482:
9447:
9445:
9444:
9439:
9437:
9436:
9425:
9419:
9412:
9411:
9403:
9399:
9398:
9392:
9388:
9383:
9374:
9373:
9346:
9344:
9343:
9338:
9336:
9335:
9324:
9318:
9311:
9309:
9301:
9280:
9278:
9277:
9272:
9270:
9269:
9258:
9252:
9245:
9243:
9235:
9215:
9211:
9203:
9201:
9200:
9195:
9193:
9192:
9181:
9174:
9172:
9163:
9162:
9153:
9152:
9131:
9129:
9128:
9123:
9121:
9120:
9109:
9102:
9100:
9099:
9094:
9082:
9081:
9076:
9064:
9059:
9051:
9030:
9028:
9027:
9022:
9020:
9019:
9008:
9001:
8999:
8991:
8971:
8955:
8953:
8952:
8947:
8945:
8944:
8933:
8926:
8924:
8912:
8911:
8906:
8904:
8899:
8898:
8896:
8895:
8894:
8885:
8877:
8862:
8857:
8856:
8844:
8843:
8833:
8809:
8807:
8806:
8801:
8799:
8798:
8787:
8780:
8778:
8771:
8766:
8760:
8759:
8754:
8748:
8743:
8742:
8734:
8730:
8729:
8724:
8723:
8718:
8712:
8707:
8698:
8697:
8681:
8679:
8678:
8673:
8671:
8670:
8659:
8652:
8644:
8634:
8632:
8631:
8626:
8624:
8623:
8612:
8605:
8600:
8599:
8597:
8589:
8587:
8585:
8577:
8575:
8570:
8567:
8551:
8549:
8548:
8543:
8541:
8540:
8529:
8522:
8520:
8519:
8517:
8509:
8507:
8505:
8497:
8488:
8478:
8476:
8475:
8470:
8468:
8467:
8456:
8449:
8444:
8443:
8438:
8433:
8428:
8425:
8404:
8402:
8401:
8396:
8394:
8393:
8382:
8375:
8371:
8370:
8368:
8367:
8365:
8364:
8359:
8358:
8349:
8340:
8339:
8330:
8321:
8320:
8311:
8298:
8297:
8292:
8288:
8286:
8269:
8252:
8236:
8234:
8233:
8228:
8226:
8225:
8214:
8207:
8205:
8204:
8199:
8190:
8180:
8178:
8177:
8172:
8170:
8169:
8158:
8151:
8146:
8145:
8140:
8135:
8130:
8127:
8117:
8115:
8114:
8109:
8107:
8106:
8095:
8088:
8080:
8059:
8057:
8056:
8051:
8049:
8048:
8037:
8030:
8028:
8027:
8025:
8024:
8022:
8021:
8016:
8015:
8006:
7997:
7996:
7987:
7978:
7977:
7968:
7956:
7951:
7947:
7946:
7938:
7933:
7925:
7920:
7912:
7907:
7899:
7875:
7873:
7872:
7867:
7865:
7864:
7853:
7846:
7841:
7840:
7835:
7830:
7825:
7822:
7817:
7812:
7807:
7802:
7790:
7788:
7787:
7782:
7780:
7779:
7768:
7761:
7760:
7755:
7751:
7750:
7748:
7747:
7731:
7726:
7721:
7720:
7687:
7685:
7684:
7679:
7677:
7676:
7664:
7662:
7654:
7644:
7642:
7641:
7636:
7634:
7633:
7621:
7619:
7618:
7609:
7608:
7599:
7578:
7576:
7575:
7570:
7568:
7567:
7555:
7546:
7545:
7540:
7532:
7507:
7505:
7504:
7499:
7497:
7496:
7484:
7482:
7480:
7475:
7467:
7459:
7438:
7436:
7435:
7430:
7428:
7427:
7416:
7409:
7400:
7399:
7383:
7381:
7380:
7375:
7373:
7372:
7361:
7354:
7349:
7344:
7339:
7329:
7327:
7326:
7321:
7319:
7318:
7307:
7300:
7295:
7290:
7285:
7275:
7273:
7272:
7267:
7265:
7264:
7252:
7250:
7242:
7231:Euler's constant
7228:
7226:
7225:
7220:
7204:
7202:
7201:
7196:
7184:
7182:
7181:
7176:
7171:
7170:
7124:
7122:
7121:
7116:
7114:
7113:
7095:
7090:
7085:
7080:
7062:that are either
7053:
7038:
7036:
7035:
7030:
7028:
7027:
7015:
7010:
7005:
7000:
6979:
6977:
6976:
6971:
6969:
6968:
6956:
6948:
6929:
6920:
6918:
6917:
6912:
6904:
6902:
6901:
6889:
6884:
6882:
6881:
6869:
6864:
6862:
6861:
6849:
6844:
6842:
6841:
6829:
6824:
6816:
6798:
6791:
6789:
6788:
6783:
6781:
6780:
6769:
6762:
6760:
6759:
6747:
6742:
6740:
6739:
6727:
6722:
6714:
6692:
6675:
6658:
6656:
6655:
6650:
6648:
6640:
6626:
6618:
6604:
6596:
6582:
6574:
6560:
6552:
6535:
6533:
6532:
6527:
6525:
6517:
6503:
6495:
6481:
6473:
6459:
6451:
6437:
6429:
6408:Tokyo University
6397:
6396:
6395:
6390:
6385:
6374:
6367:
6365:
6364:
6359:
6357:
6356:
6347:
6332:
6330:
6329:
6324:
6322:
6321:
6317:
6304:
6303:
6260:
6258:
6257:
6252:
6246:
6241:
6223:
6222:
6198:
6197:
6182:
6181:
6160:
6159:
6150:
6149:
6125:
6124:
6112:
6111:
6094:
6088:
6081:
6080:
6056:
6045:
6044:
6017:
6016:
5991:
5989:
5988:
5983:
5981:
5976:
5962:
5961:
5950:
5940:
5935:
5930:
5929:
5894:
5882:
5871:
5869:
5868:
5863:
5861:
5859:
5858:
5857:
5853:
5831:
5830:
5796:
5762:
5761:
5742:
5739:
5734:
5713:
5705:
5688:David Chudnovsky
5683:
5681:
5680:
5675:
5673:
5671:
5670:
5669:
5657:
5656:
5637:
5602:
5599:
5594:
5579:
5574:
5573:
5568:
5562:
5557:
5549:
5530:
5524:
5522:
5521:
5516:
5514:
5506:
5492:
5484:
5470:
5444:
5439:double factorial
5436:
5430:
5428:
5427:
5422:
5420:
5416:
5412:
5411:
5407:
5406:
5402:
5387:
5379:
5366:
5358:
5345:
5337:
5326:
5324:
5322:
5294:
5292:
5291:
5290:
5278:
5277:
5262:
5259:
5254:
5236:
5234:
5208:
5200:
5197:
5192:
5170:
5162:
5150:
5148:
5147:
5146:
5137:
5136:
5117:
5116:
5107:
5105:
5103:
5092:
5081:
5076:
5074:
5073:
5072:
5063:
5062:
5043:
5042:
5033:
5031:
5023:
5018:
5016:
5015:
5014:
4995:
4990:
4988:
4987:
4986:
4977:
4976:
4932:
4931:
4930:
4898:
4895:
4890:
4875:
4873:
4872:
4871:
4852:
4814:
4812:
4811:
4806:
4804:
4800:
4793:
4791:
4790:
4789:
4770:
4765:
4763:
4762:
4761:
4742:
4737:
4735:
4734:
4733:
4714:
4709:
4707:
4706:
4705:
4686:
4679:
4674:
4669:
4667:
4653:
4652:
4651:
4642:
4634:
4625:
4622:
4617:
4602:
4597:
4592:
4590:
4576:
4575:
4574:
4552:
4549:
4544:
4529:
4524:
4500:
4498:
4497:
4492:
4490:
4477:
4473:
4471:
4469:
4467:
4465:
4463:
4461:
4459:
4457:
4446:
4438:
4430:
4422:
4414:
4406:
4398:
4390:
4382:
4343:
4330:F. C. M. StΓΈrmer
4325:
4323:
4322:
4317:
4315:
4307:
4293:
4285:
4271:
4263:
4249:
4241:
4227:
4219:
4204:
4202:
4201:
4196:
4194:
4186:
4172:
4164:
4150:
4142:
4128:
4120:
4106:
4098:
4080:
4078:
4077:
4072:
4070:
4062:
4048:
4040:
4026:
4018:
4004:
3996:
3981:
3979:
3978:
3973:
3971:
3963:
3952:
3944:
3930:
3922:
3908:
3900:
3888:
3877:
3864:
3860:
3853:
3849:
3843:
3841:
3840:
3835:
3815:
3814:
3802:
3801:
3771:
3770:
3717:
3715:
3714:
3709:
3707:
3699:
3688:
3680:
3666:
3658:
3628:
3624:
3608:Willebrord Snell
3605:
3599:(circa 150 CE).
3594:
3571:
3569:
3568:
3563:
3558:
3550:
3536:
3528:
3513:
3504:
3450:
3448:
3447:
3442:
3437:
3435:
3434:
3422:
3421:
3412:
3407:
3406:
3389:
3387:
3386:
3385:
3373:
3372:
3362:
3361:
3360:
3351:
3350:
3337:
3332:
3331:
3309:
3303:
3292:
3280:
3232:
3228:
3155:2 Chronicles 4:2
3148:
3122:
3120:
3119:
3115:
3105:
3076:
3065:
3054:
3047:
3038:
3033:
3024:
3019:
3015:
3014:
3010:
3001:
3000:
2996:
2991:
2978:
2971:
2960:
2953:
2949:
2931:
2920:
2916:
2909:
2902:
2892:
2881:
2870:
2853:
2849:
2837:
2829:
2810:
2808:
2807:
2802:
2797:
2795:
2794:
2793:
2789:
2767:
2766:
2732:
2698:
2697:
2678:
2675:
2670:
2649:
2641:
2629:
2618:
2607:
2603:
2599:
2587:
2586:
2580:
2554:
2540:
2538:
2537:
2532:
2524:
2522:
2521:
2516:
2507:
2489:
2485:
2478:
2476:
2475:
2470:
2468:
2466:
2465:
2464:
2452:
2451:
2432:
2397:
2394:
2389:
2374:
2369:
2368:
2363:
2357:
2352:
2344:
2332:
2310:
2305:
2294:
2290:
2285:
2271:
2243:
2239:
2237:
2236:
2231:
2199:
2190:
2168:Gregory's series
2159:
2157:
2156:
2151:
2142:
2141:
2135:
2127:
2122:
2114:
2112:
2111:
2102:
2101:
2095:
2087:
2082:
2074:
2072:
2071:
2062:
2061:
2055:
2047:
2042:
2034:
2032:
2031:
2022:
2021:
2015:
2007:
2002:
1994:
1992:
1991:
1982:
1978:
1977:
1975:
1961:
1953:
1948:
1946:
1932:
1924:
1916:
1911:
1893:
1891:
1884:
1883:
1870:
1869:
1868:
1855:
1852:
1847:
1829:
1821:
1798:96-sided polygon
1795:
1772:
1753:
1730:
1728:
1727:
1722:
1698:
1696:
1695:
1690:
1659:
1646:JamshΔ«d al-KΔshΔ«
1641:
1637:
1630:
1628:
1627:
1622:
1620:
1618:
1608:
1607:
1594:
1587:
1586:
1576:
1571:
1569:
1555:
1554:
1553:
1534:
1523:
1515:
1510:
1502:
1497:
1489:
1475:
1454:
1453:
1450:
1447:
1441:
1419:
1415:
1414:
1401:
1399:
1398:
1393:
1391:
1387:
1380:
1378:
1377:
1376:
1357:
1352:
1350:
1349:
1348:
1329:
1324:
1322:
1308:
1295:
1290:
1285:
1283:
1269:
1268:
1267:
1258:
1250:
1241:
1238:
1233:
1218:
1213:
1208:
1206:
1192:
1191:
1190:
1168:
1165:
1160:
1145:
1140:
1123:
1121:
1120:
1115:
1107:
1102:
1100:
1064:
1062:
1061:
1056:
1054:
1050:
1043:
1035:
1030:
1022:
1017:
1009:
978:
976:
975:
970:
930:
921:Maclaurin series
889:
882:
881:
877:
872:
865:
834:
830:
829:
825:
820:
819:
815:
809:
803:
795:
794:
790:
785:
784:
778:
777:
771:
770:
764:
746:
744:
743:
738:
733:
732:
723:
709:
688:
687:
683:
667:
666:
662:
658: <
657:
654: <
653:
652:
648:
636:
592:
590:
589:
585:
566:
565:
561:
556:
537:
536:
532:
527:
514:Solomon's Temple
511:
497:
496:
492:
487:
468:
456:
447:
441:
433:
417:JamshΔ«d al-KΔshΔ«
395:
372:
365:
358:
344:
336:
208:JamshΔ«d al-KΔshΔ«
105:Area of a circle
91:
90:
87:
84:
81:
70:
54:
53:
40:
35:
28:
22:
22395:
22394:
22390:
22389:
22388:
22386:
22385:
22384:
22350:
22349:
22343:
22324:
22316:. London: BBC.
22305:
22286:
22259:
22246:
22205:
22186:
22181:
22180:
22170:
22168:
22157:
22153:
22144:
22140:
22130:
22128:
22122:
22118:
22108:
22106:
22101:
22100:
22096:
22086:
22084:
22071:
22070:
22066:
22058:
22054:
22037:
22033:
22012:
22008:
21978:
21974:
21951:10.2307/2975006
21931:
21927:
21918:
21917:
21913:
21892:
21888:
21867:
21863:
21856:
21842:
21838:
21825:
21821:
21814:
21800:
21787:
21780:
21766:
21762:
21746:
21743:
21739:
21731:1959.13/1043679
21706:
21702:
21671:
21664:
21628:
21624:
21619:Wayback Machine
21610:
21607:
21603:
21596:
21582:Gardner, Martin
21579:
21575:
21560:
21556:
21543:
21539:
21526:
21522:
21507:
21475:
21466:
21414:Euler, Leonhard
21392:Euler, Leonhard
21356:
21339:
21335:
21327:
21323:
21315:
21308:
21300:
21296:
21288:
21284:
21276:
21272:
21265:
21261:
21251:
21249:
21238:
21234:
21224:
21222:
21211:
21207:
21193:
21189:
21179:
21177:
21148:
21143:
21138:
21134:
21111:
21107:
21097:
21095:
21091:
21080:
21076:
21075:
21071:
21061:
21059:
21049:
21045:
21035:
21033:
21023:
21019:
21009:
21007:
20997:
20993:
20983:
20981:
20979:NumberWorld.org
20971:
20967:
20957:
20955:
20944:
20943:
20939:
20929:
20927:
20916:
20915:
20911:
20901:
20899:
20895:
20894:
20890:
20880:
20878:
20873:
20872:
20868:
20858:
20856:
20852:
20851:
20847:
20837:
20835:
20833:numberworld.org
20827:
20826:
20822:
20805:
20801:
20791:
20789:
20787:numberworld.org
20779:
20768:
20757:
20750:
20739:
20735:
20725:
20723:
20713:
20709:
20700:
20699:
20695:
20690:Wayback Machine
20681:
20677:
20668:
20667:
20663:
20653:
20651:
20638:
20637:
20633:
20618:10.2307/2003813
20601:
20591:
20578:
20573:10.2307/3608485
20566:
20562:
20552:
20550:
20539:
20538:
20534:
20495:
20491:
20470:
20466:
20411:
20404:
20389:
20388:
20384:
20372:
20350:
20331:
20310:
20306:
20301:
20297:
20257:
20254:
20250:
20235:
20223:
20197:
20196:
20152:
20148:
20142:
20130:
20126:
20120:
20119:
20117:
20105:
20087:
20083:
20077:
20065:
20061:
20055:
20054:
20052:
20040:
20019:
20004:
20003:
20001:
19999:
19996:
19995:
19994:
19952:
19948:
19932:
19926:
19922:
19912:
19910:
19907:
19901:
19897:
19868:
19864:
19845:
19841:
19830:
19826:
19811:
19807:
19788:
19781:
19774:
19756:
19752:
19745:
19731:
19727:
19717:
19715:
19706:
19705:
19701:
19682:
19678:
19677:
19655:
19651:
19646:Wayback Machine
19637:
19633:
19625:
19609:. Reprinted in
19580:
19576:
19557:
19553:
19536:
19532:
19487:
19483:
19474:
19470:
19463:
19442:
19438:
19429:
19425:
19414:
19410:
19402:
19396:
19392:
19383:
19378:
19355:
19351:
19343:, addressed by
19336:
19326:
19322:
19312:
19310:
19309:on 18 July 2011
19297:
19293:
19286:
19268:
19261:
19243:
19222:
19218:
19207:
19203:
19193:
19191:
19181:
19177:
19152:
19141:
19136:
19109:
19098:available from
19079:
19077:
19075:
19069:
19067:
19066:
19060:
19038:
19036:
19034:
19026:
19006:
18993:
18970:
18967:
18965:Special purpose
18949:, for instance
18938:
18927:
18920:
18918:General purpose
18903:
18900:
18896:
18887:
18881:
18876:
18836:
18832:
18800:
18797:
18796:
18762:
18758:
18735:
18732:
18731:
18671:
18668:
18667:
18616:
18612:
18597:
18593:
18568:
18533:
18531:
18525:
18514:
18500:
18496:
18491:
18478:
18476:
18473:
18472:
18450:
18418:
18414:
18408:
18404:
18394:
18359:
18357:
18351:
18340:
18323:
18319:
18317:
18304:
18302:
18299:
18298:
18253:
18249:
18239:
18232:
18228:
18218:
18211:
18207:
18197:
18163:
18156:
18152:
18143:
18139:
18138:
18136:
18130:
18119:
18084:
18076:
18074:
18068:
18057:
18040:
18038:
18035:
18034:
18027:
17992:
17987:
17968:
17962:
17958:
17956:
17937:
17931:
17927:
17925:
17906:
17900:
17896:
17894:
17875:
17869:
17865:
17863:
17844:
17839:
17820:
17814:
17810:
17808:
17804:
17800:
17789:
17785:
17779:
17765:
17764:
17762:
17756:
17745:
17733:
17729:
17724:
17716:
17713:
17712:
17702:Fabrice Bellard
17695:
17675:
17625:
17620:
17601:
17596:
17577:
17572:
17553:
17548:
17547:
17543:
17535:
17531:
17526:
17520:
17509:
17497:
17494:
17493:
17487:infinite series
17482:
17467:David H. Bailey
17447:
17432:
17429:
17392:
17387:
17368:
17362:
17358:
17356:
17337:
17331:
17327:
17325:
17306:
17300:
17296:
17294:
17275:
17269:
17265:
17263:
17244:
17239:
17220:
17214:
17210:
17208:
17204:
17200:
17189:
17185:
17178:
17174:
17164:
17162:
17156:
17145:
17133:
17129:
17124:
17116:
17113:
17112:
17105:
17101:
17098:Fabrice Bellard
17086:
17053:
17046:
17042:
17033:
17029:
17025:
17023:
17017:
17006:
16988:
16985:
16984:
16977:
16962:
16958:
16954:
16950:
16946:
16926:
16912:
16908:
16907:
16886:
16881:
16862:
16857:
16838:
16833:
16814:
16809:
16808:
16804:
16798:
16787:
16775:
16772:
16771:
16764:
16759:
16754:
16747:
16725:
16724:
16708:
16695:
16682:
16669:
16656:
16643:
16630:
16615:
16614:
16587:
16583:
16582:
16574:
16562:
16556:
16555:
16554:
16547:
16545:
16539:
16528:
16499:
16495:
16494:
16489:
16471:
16467:
16466:
16461:
16443:
16439:
16438:
16433:
16415:
16411:
16410:
16405:
16396:
16395:
16374:
16370:
16345:
16328:
16326:
16314:
16310:
16291:
16280:
16278:
16266:
16262:
16249:
16244:
16231:
16230:
16226:
16206:
16202:
16190:
16186:
16176:
16169:
16167:
16164:
16163:
16136:
16119:
16115:
16107:
16104:
16103:
16097:
16076:
16073:
16072:
16047:
16043:
16041:
16038:
16037:
16012:
16008:
15995:
15991:
15982:
15964:
15947:
15945:
15942:
15941:
15908:
15889:
15870:
15851:
15825:
15821:
15816:
15804:
15793:
15776:
15774:
15771:
15770:
15751:
15748:
15747:
15728:
15725:
15724:
15708:
15705:
15704:
15688:
15685:
15684:
15683:to approximate
15667:
15666:
15655:
15634:
15630:
15613:
15609:
15604:
15594:
15590:
15568:
15564:
15546:
15542:
15538:
15537:
15526:
15505:
15501:
15484:
15480:
15475:
15465:
15461:
15439:
15435:
15417:
15413:
15409:
15408:
15384:
15380:
15376:
15375:
15364:
15343:
15339:
15334:
15332:
15329:
15328:
15293:
15287:
15282:
15264:
15258:
15253:
15248:
15236:
15229:
15224:
15223:
15222:
15221:
15219:
15203:
15198:
15197:
15191:
15180:
15153:
15150:
15149:
15114:
15110:
15104:
15100:
15090:
15075:
15071:
15069:
15043:
15036:
15032:
15014:
15010:
15009:
15007:
15001:
14990:
14966:
14963:
14962:
14937:
14928:
14924:
14922:
14919:
14918:
14893:
14889:
14881:
14872:
14868:
14866:
14863:
14862:
14823:
14819:
14806:
14802:
14793:
14770:
14766:
14761:
14759:
14756:
14755:
14736:
14733:
14732:
14684:
14680:
14670:
14663:
14659:
14649:
14642:
14638:
14628:
14614:
14611:
14610:
14590:
14589:
14573:
14560:
14547:
14534:
14521:
14508:
14495:
14482:
14469:
14454:
14453:
14423:
14411:
14405:
14404:
14403:
14402:
14397:
14382:
14378:
14377:
14372:
14370:
14364:
14353:
14319:
14314:
14305:
14301:
14286:
14282:
14281:
14276:
14274:
14268:
14257:
14220:
14215:
14205:
14200:
14198:
14192:
14181:
14165:
14164:
14120:
14115:
14084:
14079:
14077:
14047:
14042:
14017:
14012:
14010:
13986:
13981:
13962:
13957:
13955:
13937:
13932:
13919:
13914:
13912:
13900:
13895:
13888:
13883:
13881:
13874:
13870:
13860:
13853:
13851:
13848:
13847:
13840:
13837:
13831:
13806:
13803:
13802:
13786:
13783:
13782:
13776:
13734:
13721:
13708:
13695:
13694:
13690:
13666:
13661:
13652:
13648:
13638:
13633:
13631:
13625:
13614:
13599:
13596:
13595:
13586:
13581:
13567:
13563:
13562:
13557:
13553:
13552:
13519:
13515:
13511:
13498:
13494:
13493:
13491:
13469:
13463:
13459:
13457:
13447:
13441:
13437:
13435:
13425:
13420:
13418:
13415:
13414:
13330:
13325:
13316:
13312:
13311:
13306:
13304:
13294:
13289:
13280:
13276:
13275:
13270:
13268:
13258:
13253:
13244:
13240:
13239:
13234:
13232:
13222:
13217:
13208:
13204:
13191:
13186:
13184:
13165:
13152:
13148:
13138:
13136:
13130:
13119:
13104:
13101:
13100:
13044:
13039:
13030:
13026:
13025:
13020:
13018:
13011:
13006:
12997:
12993:
12992:
12987:
12985:
12978:
12973:
12964:
12960:
12959:
12954:
12952:
12945:
12940:
12931:
12927:
12926:
12921:
12919:
12912:
12907:
12898:
12894:
12893:
12888:
12886:
12842:
12837:
12828:
12824:
12823:
12818:
12816:
12809:
12804:
12795:
12791:
12790:
12785:
12783:
12776:
12771:
12762:
12758:
12757:
12752:
12750:
12743:
12738:
12729:
12725:
12724:
12719:
12717:
12710:
12705:
12698:
12693:
12691:
12683:
12680:
12679:
12640:
12635:
12626:
12622:
12621:
12616:
12614:
12607:
12602:
12593:
12589:
12588:
12583:
12581:
12574:
12569:
12560:
12556:
12555:
12550:
12548:
12541:
12533:
12530:
12529:
12518:
12511:
12502:
12398:
12382:
12377:
12373:
12372:
12368:
11948:
11944:
11940:
11936:
11931:
11926:
11922:
11916:
11910:
11904:
11898:
11894:
11874:
11873:
11856:
11852:
11843:
11839:
11837:
11832:
11830:
11824:
11823:
11809:
11805:
11796:
11792:
11790:
11785:
11783:
11773:
11772:
11766:
11752:
11741:
11727:
11715:
11711:
11706:
11694:
11682:
11679:
11678:
11668:
11664:
11633:
11629:
11620:
11616:
11614:
11612:
11609:
11608:
11599:
11593:
11589:
11585:
11575:
11569:
11563:
11557:
11532:
11528:
11519:
11515:
11513:
11505:
11502:
11501:
11492:
11486:
11477:
11471:
11447:
11443:
11432:
11429:
11428:
11417:
11413:
11406:
11396:
11375:
11373:
11370:
11369:
11344:
11331:
11318:
11305:
11292:
11279:
11266:
11253:
11240:
11227:
11214:
11201:
11199:
11196:
11195:
11185:
11177:
11142:
11138:
11136:
11125:
11122:
11121:
11090:
11088:
11085:
11084:
11040:
11032:
11029:
11028:
10998:
10997:
10987:
10963:
10956:
10950:
10949:
10939:
10920:
10914:
10913:
10903:
10890:
10872:
10865:
10859:
10858:
10848:
10824:
10817:
10810:
10808:
10805:
10804:
10770:
10766:
10764:
10738:
10734:
10732:
10717:
10713:
10699:
10695:
10693:
10678:
10674:
10660:
10656:
10654:
10637:
10634:
10633:
10598:
10597:
10585:
10581:
10566:
10565:
10558:
10556:
10554:
10551:
10550:
10536:
10530:
10505:
10498:
10491:
10490:
10474:
10470:
10454:
10441:
10431:
10418:
10408:
10395:
10374:
10370:
10360:
10358:
10356:
10353:
10352:
10295:
10291:
10284:
10266:
10262:
10252:
10250:
10248:
10245:
10244:
10208:Heegner numbers
10182:
10178:
10122:
10118:
10108:
10106:
10104:
10101:
10100:
10085:
10079:
10058:
10054:
10001:
9997:
9984:
9979:
9975:
9969:
9965:
9963:
9962:
9958:
9942:
9940:
9937:
9936:
9906:
9902:
9862:
9858:
9856:
9854:
9851:
9850:
9820:
9816:
9782:
9778:
9776:
9774:
9771:
9770:
9753:
9732:
9728:
9698:
9683:
9667:
9663:
9661:
9646:
9645:
9641:
9640:
9638:
9635:
9634:
9605:
9601:
9570:
9568:
9565:
9564:
9545:
9541:
9507:
9503:
9501:
9499:
9496:
9495:
9480:
9473:
9466:
9459:
9453:
9432:
9428:
9404:
9387:
9372:
9371:
9367:
9366:
9364:
9361:
9360:
9331:
9327:
9305:
9300:
9298:
9295:
9294:
9265:
9261:
9239:
9234:
9232:
9229:
9228:
9213:
9209:
9188:
9184:
9168:
9158:
9154:
9151:
9149:
9146:
9145:
9116:
9112:
9093:
9083:
9075:
9065:
9063:
9050:
9048:
9045:
9044:
9015:
9011:
8995:
8990:
8988:
8985:
8984:
8969:
8940:
8936:
8920:
8910:
8900:
8890:
8886:
8876:
8866:
8861:
8852:
8848:
8839:
8835:
8834:
8832:
8830:
8827:
8826:
8794:
8790:
8765:
8761:
8753:
8749:
8747:
8735:
8717:
8713:
8711:
8696:
8695:
8691:
8690:
8688:
8685:
8684:
8666:
8662:
8643:
8641:
8638:
8637:
8619:
8615:
8593:
8588:
8581:
8576:
8569:
8568:
8566:
8558:
8555:
8554:
8536:
8532:
8513:
8508:
8501:
8496:
8492:
8487:
8485:
8482:
8481:
8463:
8459:
8437:
8427:
8426:
8424:
8422:
8419:
8418:
8389:
8385:
8354:
8350:
8348:
8341:
8335:
8331:
8329:
8322:
8316:
8312:
8310:
8303:
8299:
8293:
8270:
8253:
8251:
8247:
8246:
8244:
8241:
8240:
8221:
8217:
8198:
8194:
8189:
8187:
8184:
8183:
8165:
8161:
8139:
8129:
8128:
8126:
8124:
8121:
8120:
8102:
8098:
8079:
8077:
8074:
8073:
8044:
8040:
8011:
8007:
8005:
7998:
7992:
7988:
7986:
7979:
7973:
7969:
7967:
7960:
7955:
7937:
7924:
7911:
7898:
7891:
7887:
7882:
7879:
7878:
7860:
7856:
7834:
7824:
7823:
7821:
7811:
7801:
7799:
7796:
7795:
7775:
7771:
7756:
7743:
7739:
7725:
7719:
7712:
7708:
7707:
7705:
7702:
7701:
7672:
7668:
7658:
7653:
7651:
7648:
7647:
7629:
7625:
7614:
7610:
7604:
7600:
7598:
7596:
7593:
7592:
7563:
7559:
7544:
7531:
7529:
7526:
7525:
7492:
7488:
7474:
7466:
7458:
7456:
7453:
7452:
7423:
7419:
7398:
7390:
7387:
7386:
7368:
7364:
7348:
7338:
7336:
7333:
7332:
7314:
7310:
7294:
7284:
7282:
7279:
7278:
7260:
7256:
7246:
7241:
7239:
7236:
7235:
7214:
7211:
7210:
7190:
7187:
7186:
7166:
7162:
7142:
7139:
7138:
7109:
7105:
7089:
7079:
7077:
7074:
7073:
7060:right triangles
7051:
7023:
7019:
7009:
6999:
6997:
6994:
6993:
6964:
6960:
6947:
6945:
6942:
6941:
6927:
6897:
6893:
6888:
6877:
6873:
6868:
6857:
6853:
6848:
6837:
6833:
6828:
6815:
6804:
6801:
6800:
6796:
6776:
6772:
6755:
6751:
6746:
6735:
6731:
6726:
6713:
6705:
6702:
6701:
6696:
6690:
6683:
6673:
6639:
6617:
6595:
6573:
6551:
6549:
6546:
6545:
6516:
6494:
6472:
6450:
6428:
6426:
6423:
6422:
6404:Yasumasa Kanada
6393:
6392:
6388:
6387:
6383:
6372:
6352:
6348:
6343:
6338:
6335:
6334:
6333:, the sequence
6313:
6309:
6305:
6299:
6295:
6269:
6266:
6265:
6242:
6231:
6212:
6208:
6187:
6183:
6168:
6164:
6155:
6151:
6139:
6135:
6120:
6116:
6101:
6097:
6076:
6072:
6052:
6040:
6036:
6006:
6002:
6000:
5997:
5996:
5975:
5957:
5953:
5934:
5925:
5921:
5919:
5916:
5915:
5892:
5889:
5880:
5849:
5836:
5832:
5826:
5822:
5797:
5757:
5753:
5743:
5741:
5735:
5724:
5704:
5702:
5699:
5698:
5662:
5658:
5652:
5648:
5638:
5603:
5601:
5595:
5584:
5567:
5563:
5561:
5548:
5546:
5543:
5542:
5528:
5505:
5483:
5466:
5464:
5461:
5460:
5442:
5434:
5418:
5417:
5392:
5388:
5378:
5371:
5367:
5357:
5350:
5346:
5336:
5300:
5295:
5286:
5282:
5273:
5269:
5268:
5263:
5261:
5255:
5244:
5209:
5201:
5199:
5193:
5182:
5171:
5161:
5158:
5157:
5142:
5138:
5132:
5128:
5118:
5112:
5108:
5106:
5093:
5082:
5080:
5068:
5064:
5058:
5054:
5044:
5038:
5034:
5032:
5022:
5010:
5006:
4999:
4994:
4982:
4978:
4972:
4968:
4933:
4923:
4919:
4899:
4897:
4891:
4880:
4867:
4863:
4856:
4851:
4844:
4831:
4829:
4826:
4825:
4785:
4781:
4774:
4769:
4757:
4753:
4746:
4741:
4729:
4725:
4718:
4713:
4701:
4697:
4690:
4685:
4684:
4680:
4673:
4654:
4647:
4643:
4633:
4626:
4624:
4618:
4607:
4596:
4577:
4567:
4563:
4553:
4551:
4545:
4534:
4523:
4515:
4512:
4511:
4488:
4487:
4475:
4474:
4445:
4437:
4429:
4421:
4413:
4405:
4397:
4389:
4381:
4371:
4364:
4362:
4359:
4358:
4352:Viète's formula
4341:
4338:
4306:
4284:
4262:
4240:
4218:
4216:
4213:
4212:
4185:
4163:
4141:
4119:
4097:
4095:
4092:
4091:
4061:
4039:
4017:
3995:
3993:
3990:
3989:
3962:
3943:
3921:
3899:
3897:
3894:
3893:
3886:
3875:
3862:
3858:
3851:
3847:
3810:
3806:
3797:
3793:
3766:
3762:
3748:
3745:
3744:
3739:complex numbers
3698:
3679:
3657:
3655:
3652:
3651:
3641:
3635:
3626:
3622:
3614:, published by
3612:Viète's formula
3603:
3592:
3579:reports in his
3549:
3527:
3522:
3519:
3518:
3512:
3506:
3503:
3496:
3489:
3482:
3475:
3468:
3461:
3455:
3427:
3423:
3417:
3413:
3411:
3399:
3395:
3381:
3377:
3368:
3364:
3363:
3356:
3352:
3346:
3342:
3338:
3336:
3324:
3320:
3318:
3315:
3314:
3305:
3302:
3294:
3291:
3283:
3278:
3271:
3266:
3260:
3230:
3226:
3146:
3139:
3133:
3117:
3113:
3112:
3107:
3103:
3089:
3087:Indiana Pi Bill
3083:
3071:
3060:
3057:Indiana Pi Bill
3052:
3049:
3045:
3012:
3008:
3007:
2998:
2994:
2993:
2989:
2986:
2976:
2969:
2958:
2951:
2947:
2929:
2918:
2914:
2907:
2900:
2890:
2879:
2876:Fabrice Bellard
2868:
2860:
2851:
2847:
2840:Yasumasa Kanada
2835:
2827:
2820:Yasumasa Kanada
2785:
2772:
2768:
2762:
2758:
2733:
2693:
2689:
2679:
2677:
2671:
2660:
2640:
2638:
2635:
2634:
2627:
2616:
2605:
2601:
2597:
2584:
2582:
2578:
2552:
2515:
2511:
2506:
2498:
2495:
2494:
2487:
2483:
2457:
2453:
2447:
2443:
2433:
2398:
2396:
2390:
2379:
2362:
2358:
2356:
2343:
2341:
2338:
2337:
2330:
2323:
2317:
2308:
2303:
2292:
2283:
2281:
2269:
2250:Thomas de Lagny
2241:
2188:
2186:
2183:
2182:
2137:
2136:
2126:
2113:
2107:
2106:
2097:
2096:
2086:
2073:
2067:
2066:
2057:
2056:
2046:
2033:
2027:
2026:
2017:
2016:
2006:
1993:
1987:
1986:
1962:
1954:
1952:
1933:
1925:
1923:
1922:
1918:
1912:
1901:
1879:
1875:
1871:
1864:
1860:
1856:
1854:
1848:
1837:
1820:
1818:
1815:
1814:
1793:
1770:
1756:Viète's formula
1751:
1744:
1736:regular polygon
1710:
1707:
1706:
1672:
1669:
1668:
1657:
1639:
1635:
1603:
1599:
1595:
1582:
1578:
1577:
1575:
1556:
1549:
1545:
1535:
1533:
1514:
1501:
1488:
1471:
1466:
1463:
1462:
1451:
1448:
1445:
1443:
1439:
1425:
1417:
1412:
1410:
1372:
1368:
1361:
1356:
1344:
1340:
1333:
1328:
1312:
1307:
1300:
1296:
1289:
1270:
1263:
1259:
1249:
1242:
1240:
1234:
1223:
1212:
1193:
1183:
1179:
1169:
1167:
1161:
1150:
1139:
1131:
1128:
1127:
1101:
1096:
1073:
1070:
1069:
1034:
1021:
1008:
1001:
997:
986:
983:
982:
940:
937:
936:
928:
925:infinite series
909:
879:
875:
874:
870:
866:
859:
841:Gupta-era India
832:
827:
823:
822:
817:
813:
812:
807:
801:
792:
788:
787:
782:
780:
775:
773:
768:
766:
762:
728:
724:
719:
705:
694:
691:
690:
685:
681:
680:
679:used the value
664:
660:
659:
655:
650:
646:
645:
637:
624:
617:
615:
613:
609:
587:
583:
582:
581:
563:
559:
558:
554:
534:
530:
529:
525:
509:
494:
490:
489:
485:
466:
463:
454:
445:
439:
431:
393:
376:
342:
334:
302:Indiana pi bill
285:A History of Pi
263:Yasumasa Kanada
88:
85:
82:
79:
77:
68:
42:
38:
33:
26:
20:
17:
12:
11:
5:
22393:
22383:
22382:
22377:
22372:
22367:
22362:
22360:Approximations
22348:
22347:
22341:
22328:
22322:
22309:
22303:
22290:
22284:
22271:
22257:
22240:Beckmann, Petr
22236:
22201:(April 1997).
22199:Plouffe, Simon
22185:
22182:
22179:
22178:
22151:
22147:PiFast timings
22138:
22116:
22094:
22064:
22052:
22031:
22006:
21972:
21945:(3): 195β203.
21925:
21911:
21886:
21861:
21854:
21836:
21826:Hoffman, D.W.
21819:
21812:
21785:
21778:
21760:
21737:
21716:(8): 681β687.
21700:
21681:(2): 115β126.
21662:
21622:
21601:
21594:
21573:
21554:
21537:
21520:
21505:
21464:
21354:"Estimating Ο"
21333:
21321:
21306:
21294:
21282:
21270:
21259:
21232:
21205:
21187:
21132:
21121:(3): 136β140.
21105:
21069:
21043:
21017:
20991:
20965:
20937:
20909:
20888:
20877:. 26 June 2019
20866:
20845:
20820:
20799:
20766:
20748:
20733:
20707:
20693:
20675:
20661:
20631:
20576:
20560:
20532:
20489:
20464:
20402:
20399:. 11 May 2015.
20396:Stack Exchange
20382:
20348:
20315:. Supplement.
20309:, exprimΓ©e en
20295:
20248:
20198:3.14159, &
20185:
20182:
20179:
20176:
20172:
20169:
20164:
20155:
20151:
20147:
20141:
20133:
20129:
20125:
20113:
20110:
20104:
20099:
20090:
20086:
20082:
20076:
20068:
20064:
20060:
20048:
20045:
20039:
20034:
20027:
20024:
20018:
20012:
20009:
19981:of particular
19955:Jones, William
19946:
19920:
19905:"Digits of Pi"
19895:
19862:
19839:
19824:
19815:Ganita Bharati
19805:
19779:
19772:
19766:. p. 58.
19750:
19743:
19725:
19699:
19675:
19674:
19671:
19649:
19631:
19624:978-0387205717
19623:
19590:(4): 325β340,
19574:
19551:
19530:
19481:
19468:
19461:
19436:
19423:
19408:
19390:
19377:978-0871692061
19376:
19370:. p. 78.
19349:
19345:Rabbi Nehemiah
19320:
19291:
19284:
19270:Rossi (2007).
19259:
19241:
19216:
19201:
19175:
19138:
19137:
19135:
19132:
19131:
19130:
19125:
19120:
19115:
19108:
19105:
19104:
19103:
19090:
19053:
19012:
18989:
18977:and efficient
18966:
18963:
18926:can calculate
18919:
18916:
18899:
18893:
18880:
18877:
18875:
18872:
18869:
18868:
18865:
18862:
18856:
18855:
18844:
18839:
18835:
18831:
18828:
18825:
18822:
18819:
18816:
18813:
18810:
18807:
18804:
18794:
18792:
18782:
18781:
18770:
18765:
18761:
18757:
18754:
18751:
18748:
18745:
18742:
18739:
18729:
18726:
18720:
18719:
18708:
18705:
18702:
18699:
18696:
18693:
18690:
18687:
18684:
18681:
18678:
18675:
18665:
18662:
18656:
18655:
18652:
18649:
18640:
18639:
18622:
18619:
18615:
18611:
18608:
18605:
18600:
18596:
18592:
18589:
18586:
18583:
18580:
18577:
18574:
18571:
18566:
18563:
18560:
18557:
18554:
18551:
18548:
18545:
18542:
18539:
18536:
18528:
18523:
18520:
18517:
18513:
18504:
18499:
18495:
18490:
18485:
18482:
18447:
18446:
18440:
18424:
18421:
18417:
18411:
18407:
18403:
18400:
18397:
18392:
18389:
18386:
18383:
18380:
18377:
18374:
18371:
18368:
18365:
18362:
18354:
18349:
18346:
18343:
18339:
18333:
18327:
18322:
18316:
18311:
18308:
18296:
18294:
18288:
18276:
18271:
18266:
18262:
18259:
18256:
18252:
18246:
18243:
18238:
18235:
18231:
18225:
18222:
18217:
18214:
18210:
18204:
18201:
18196:
18193:
18190:
18184:
18181:
18178:
18175:
18172:
18169:
18166:
18159:
18155:
18151:
18146:
18142:
18133:
18128:
18125:
18122:
18118:
18114:
18108:
18105:
18102:
18099:
18096:
18093:
18090:
18087:
18082:
18079:
18071:
18066:
18063:
18060:
18056:
18052:
18047:
18044:
18024:
18023:
18011:
18004:
18001:
17998:
17995:
17991:
17986:
17980:
17977:
17974:
17971:
17965:
17961:
17955:
17949:
17946:
17943:
17940:
17934:
17930:
17924:
17918:
17915:
17912:
17909:
17903:
17899:
17893:
17887:
17884:
17881:
17878:
17872:
17868:
17862:
17856:
17853:
17850:
17847:
17843:
17838:
17832:
17829:
17826:
17823:
17817:
17813:
17807:
17803:
17795:
17792:
17788:
17782:
17777:
17774:
17771:
17768:
17759:
17754:
17751:
17748:
17744:
17736:
17732:
17728:
17723:
17720:
17660:
17659:
17648:
17644:
17637:
17634:
17631:
17628:
17624:
17619:
17613:
17610:
17607:
17604:
17600:
17595:
17589:
17586:
17583:
17580:
17576:
17571:
17565:
17562:
17559:
17556:
17552:
17546:
17538:
17534:
17530:
17523:
17518:
17515:
17512:
17508:
17504:
17501:
17428:
17425:
17424:
17423:
17411:
17404:
17401:
17398:
17395:
17391:
17386:
17380:
17377:
17374:
17371:
17365:
17361:
17355:
17349:
17346:
17343:
17340:
17334:
17330:
17324:
17318:
17315:
17312:
17309:
17303:
17299:
17293:
17287:
17284:
17281:
17278:
17272:
17268:
17262:
17256:
17253:
17250:
17247:
17243:
17238:
17232:
17229:
17226:
17223:
17217:
17213:
17207:
17203:
17195:
17192:
17188:
17181:
17177:
17173:
17170:
17167:
17159:
17154:
17151:
17148:
17144:
17136:
17132:
17128:
17123:
17120:
17083:
17082:
17068:
17065:
17062:
17059:
17056:
17049:
17045:
17041:
17036:
17032:
17028:
17020:
17015:
17012:
17009:
17005:
17001:
16998:
16995:
16992:
16943:
16942:
16929:
16924:
16919:
16916:
16911:
16905:
16898:
16895:
16892:
16889:
16885:
16880:
16874:
16871:
16868:
16865:
16861:
16856:
16850:
16847:
16844:
16841:
16837:
16832:
16826:
16823:
16820:
16817:
16813:
16807:
16801:
16796:
16793:
16790:
16786:
16782:
16779:
16746:
16743:
16739:
16738:
16723:
16720:
16715:
16712:
16707:
16702:
16699:
16694:
16689:
16686:
16681:
16676:
16673:
16668:
16663:
16660:
16655:
16650:
16647:
16642:
16637:
16634:
16629:
16626:
16623:
16620:
16618:
16616:
16610:
16607:
16604:
16601:
16598:
16595:
16590:
16586:
16577:
16572:
16568:
16565:
16559:
16553:
16550:
16542:
16537:
16534:
16531:
16527:
16523:
16519:
16516:
16510:
16507:
16502:
16498:
16493:
16488:
16482:
16479:
16474:
16470:
16465:
16460:
16454:
16451:
16446:
16442:
16437:
16432:
16426:
16423:
16418:
16414:
16409:
16404:
16401:
16399:
16397:
16393:
16388:
16385:
16377:
16373:
16369:
16366:
16363:
16360:
16357:
16354:
16351:
16348:
16343:
16340:
16337:
16334:
16331:
16325:
16317:
16313:
16309:
16306:
16303:
16300:
16297:
16294:
16289:
16286:
16283:
16277:
16269:
16265:
16261:
16258:
16255:
16252:
16248:
16243:
16238:
16235:
16229:
16225:
16222:
16218:
16213:
16210:
16205:
16201:
16196:
16193:
16189:
16185:
16182:
16179:
16177:
16175:
16172:
16171:
16157:
16156:
16143:
16140:
16135:
16131:
16126:
16123:
16118:
16114:
16111:
16096:
16093:
16080:
16050:
16046:
16034:
16033:
16022:
16015:
16011:
16004:
16001:
15998:
15994:
15990:
15987:
15981:
15978:
15973:
15970:
15967:
15963:
15959:
15954:
15951:
15935:
15934:
15923:
15920:
15915:
15912:
15907:
15904:
15901:
15896:
15893:
15888:
15885:
15882:
15877:
15874:
15869:
15866:
15863:
15858:
15855:
15850:
15847:
15844:
15837:
15834:
15831:
15828:
15824:
15820:
15815:
15812:
15807:
15802:
15799:
15796:
15792:
15788:
15783:
15780:
15755:
15732:
15712:
15692:
15681:
15680:
15665:
15662:
15658:
15654:
15651:
15648:
15643:
15640:
15637:
15633:
15629:
15626:
15622:
15616:
15612:
15607:
15603:
15600:
15597:
15593:
15588:
15585:
15582:
15577:
15574:
15571:
15567:
15563:
15560:
15557:
15554:
15549:
15545:
15541:
15539:
15536:
15533:
15529:
15525:
15522:
15519:
15514:
15511:
15508:
15504:
15500:
15497:
15493:
15487:
15483:
15478:
15474:
15471:
15468:
15464:
15459:
15456:
15453:
15448:
15445:
15442:
15438:
15434:
15431:
15428:
15425:
15420:
15416:
15412:
15410:
15407:
15404:
15401:
15398:
15395:
15392:
15387:
15383:
15379:
15377:
15374:
15371:
15367:
15363:
15360:
15357:
15354:
15351:
15346:
15342:
15338:
15336:
15322:
15321:
15310:
15302:
15299:
15296:
15290:
15285:
15281:
15277:
15273:
15270:
15267:
15261:
15256:
15252:
15245:
15242:
15239:
15232:
15227:
15215:
15212:
15209:
15206:
15202:
15194:
15189:
15186:
15183:
15179:
15175:
15172:
15169:
15166:
15163:
15160:
15157:
15143:
15142:
15131:
15123:
15120:
15117:
15113:
15107:
15103:
15099:
15096:
15093:
15087:
15084:
15081:
15078:
15074:
15064:
15061:
15058:
15055:
15052:
15049:
15046:
15039:
15035:
15031:
15028:
15025:
15020:
15017:
15013:
15004:
14999:
14996:
14993:
14989:
14985:
14982:
14979:
14976:
14973:
14970:
14941:
14936:
14931:
14927:
14902:
14899:
14896:
14892:
14888:
14885:
14880:
14875:
14871:
14859:
14858:
14847:
14844:
14841:
14838:
14833:
14826:
14822:
14815:
14812:
14809:
14805:
14801:
14798:
14792:
14789:
14786:
14779:
14776:
14773:
14769:
14765:
14740:
14729:
14728:
14726:
14719:
14707:
14702:
14697:
14693:
14690:
14687:
14683:
14677:
14674:
14669:
14666:
14662:
14656:
14653:
14648:
14645:
14641:
14635:
14632:
14627:
14624:
14621:
14618:
14604:
14603:
14588:
14585:
14580:
14577:
14572:
14567:
14564:
14559:
14554:
14551:
14546:
14541:
14538:
14533:
14528:
14525:
14520:
14515:
14512:
14507:
14502:
14499:
14494:
14489:
14486:
14481:
14476:
14473:
14468:
14465:
14462:
14459:
14457:
14455:
14447:
14444:
14441:
14438:
14435:
14432:
14426:
14421:
14417:
14414:
14408:
14391:
14388:
14385:
14381:
14367:
14362:
14359:
14356:
14352:
14348:
14340:
14337:
14334:
14331:
14328:
14325:
14322:
14308:
14304:
14300:
14295:
14292:
14289:
14285:
14271:
14266:
14263:
14260:
14256:
14252:
14244:
14241:
14238:
14235:
14232:
14229:
14226:
14223:
14211:
14208:
14195:
14190:
14187:
14184:
14180:
14176:
14173:
14170:
14168:
14166:
14162:
14158:
14155:
14147:
14144:
14141:
14138:
14135:
14132:
14129:
14126:
14123:
14111:
14108:
14105:
14102:
14099:
14096:
14093:
14090:
14087:
14076:
14068:
14065:
14062:
14059:
14056:
14053:
14050:
14038:
14035:
14032:
14029:
14026:
14023:
14020:
14009:
14001:
13998:
13995:
13992:
13989:
13977:
13974:
13971:
13968:
13965:
13954:
13946:
13943:
13940:
13928:
13925:
13922:
13911:
13903:
13891:
13880:
13877:
13873:
13869:
13866:
13863:
13861:
13859:
13856:
13855:
13830:
13827:
13810:
13790:
13769:
13768:
13756:
13752:
13749:
13746:
13741:
13738:
13733:
13728:
13725:
13720:
13715:
13712:
13707:
13702:
13699:
13693:
13689:
13686:
13678:
13675:
13672:
13669:
13655:
13651:
13647:
13644:
13641:
13628:
13623:
13620:
13617:
13613:
13609:
13606:
13603:
13585:
13582:
13580:
13577:
13549:
13548:
13533:
13530:
13527:
13522:
13518:
13514:
13509:
13506:
13501:
13497:
13490:
13487:
13475:
13472:
13466:
13462:
13456:
13453:
13450:
13444:
13440:
13434:
13431:
13428:
13424:
13408:
13407:
13396:
13393:
13390:
13387:
13384:
13381:
13378:
13375:
13372:
13369:
13366:
13342:
13339:
13336:
13333:
13319:
13315:
13303:
13300:
13297:
13283:
13279:
13267:
13264:
13261:
13247:
13243:
13231:
13228:
13225:
13211:
13207:
13203:
13200:
13197:
13194:
13183:
13177:
13174:
13171:
13168:
13161:
13158:
13155:
13151:
13147:
13144:
13141:
13133:
13128:
13125:
13122:
13118:
13114:
13111:
13108:
13090:
13089:
13053:
13050:
13047:
13033:
13029:
13017:
13014:
13000:
12996:
12984:
12981:
12967:
12963:
12951:
12948:
12934:
12930:
12918:
12915:
12901:
12897:
12885:
12882:
12879:
12851:
12848:
12845:
12831:
12827:
12815:
12812:
12798:
12794:
12782:
12779:
12765:
12761:
12749:
12746:
12732:
12728:
12716:
12713:
12701:
12690:
12687:
12677:
12649:
12646:
12643:
12629:
12625:
12613:
12610:
12596:
12592:
12580:
12577:
12563:
12559:
12547:
12544:
12540:
12537:
12510:
12507:
12490:
12489:
12486:
12483:
12479:
12478:
12475:
12472:
12468:
12467:
12464:
12461:
12457:
12456:
12453:
12450:
12446:
12445:
12442:
12439:
12435:
12434:
12431:
12428:
12424:
12423:
12420:
12417:
12413:
12412:
12409:
12406:
12402:
12401:
12395:
12392:
12345:
12344:
12341:
12340:
12337:
12334:
12331:
12328:
12325:
12322:
12319:
12316:
12313:
12310:
12306:
12305:
12302:
12299:
12296:
12293:
12290:
12287:
12284:
12281:
12278:
12275:
12271:
12270:
12267:
12264:
12261:
12258:
12255:
12252:
12249:
12246:
12243:
12240:
12236:
12235:
12232:
12229:
12226:
12223:
12220:
12217:
12214:
12211:
12208:
12205:
12201:
12200:
12197:
12194:
12191:
12188:
12185:
12182:
12179:
12176:
12173:
12170:
12166:
12165:
12162:
12159:
12156:
12153:
12150:
12147:
12144:
12141:
12138:
12135:
12131:
12130:
12127:
12124:
12121:
12118:
12115:
12112:
12109:
12106:
12103:
12100:
12096:
12095:
12092:
12089:
12086:
12083:
12080:
12077:
12074:
12071:
12068:
12065:
12061:
12060:
12057:
12054:
12051:
12048:
12045:
12042:
12039:
12036:
12033:
12030:
12026:
12025:
12022:
12019:
12016:
12013:
12010:
12007:
12004:
12001:
11998:
11995:
11991:
11990:
11987:
11984:
11981:
11978:
11975:
11972:
11969:
11966:
11963:
11960:
11891:
11890:
11877:
11872:
11869:
11866:
11859:
11855:
11851:
11846:
11842:
11831:
11829:
11826:
11825:
11822:
11819:
11812:
11808:
11804:
11799:
11795:
11784:
11782:
11779:
11778:
11776:
11769:
11764:
11761:
11758:
11755:
11751:
11744:
11739:
11736:
11733:
11730:
11726:
11718:
11714:
11710:
11703:
11700:
11697:
11693:
11689:
11686:
11661:
11660:
11649:
11646:
11643:
11636:
11632:
11628:
11623:
11619:
11554:
11553:
11542:
11535:
11531:
11527:
11522:
11518:
11512:
11509:
11467:
11466:
11455:
11450:
11446:
11442:
11439:
11436:
11405:
11402:
11401:
11400:
11382:
11379:
11366:
11365:
11364:
11351:
11348:
11343:
11338:
11335:
11330:
11325:
11322:
11317:
11312:
11309:
11304:
11299:
11296:
11291:
11286:
11283:
11278:
11273:
11270:
11265:
11260:
11257:
11252:
11247:
11244:
11239:
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11231:
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11221:
11218:
11213:
11208:
11205:
11190:
11189:
11169:
11168:
11167:
11166:
11152:
11146:
11141:
11135:
11132:
11129:
11119:
11108:
11105:
11102:
11097:
11094:
11079:
11078:
11074:
11073:
11047:
11044:
11039:
11036:
11013:
11012:
11011:
10996:
10991:
10986:
10983:
10980:
10977:
10971:
10968:
10962:
10959:
10957:
10955:
10952:
10951:
10948:
10943:
10938:
10935:
10932:
10929:
10926:
10923:
10921:
10919:
10916:
10915:
10912:
10907:
10902:
10899:
10894:
10889:
10886:
10880:
10877:
10871:
10868:
10866:
10864:
10861:
10860:
10857:
10852:
10847:
10844:
10841:
10838:
10832:
10829:
10823:
10820:
10818:
10816:
10813:
10812:
10799:
10798:
10797:
10786:
10781:
10778:
10773:
10769:
10763:
10760:
10757:
10754:
10749:
10746:
10741:
10737:
10731:
10728:
10725:
10720:
10716:
10710:
10707:
10702:
10698:
10692:
10689:
10686:
10681:
10677:
10671:
10668:
10663:
10659:
10653:
10650:
10647:
10644:
10641:
10624:
10623:
10622:
10608:
10601:
10596:
10593:
10588:
10584:
10580:
10577:
10574:
10569:
10564:
10561:
10545:
10544:
10540:
10539:
10534:
10527:
10526:
10525:
10509:
10502:
10495:
10488:
10485:
10482:
10477:
10473:
10469:
10466:
10463:
10458:
10453:
10450:
10445:
10440:
10435:
10430:
10427:
10422:
10417:
10412:
10407:
10404:
10399:
10394:
10391:
10388:
10385:
10380:
10377:
10373:
10369:
10366:
10363:
10347:
10346:
10342:
10341:
10330:
10329:
10328:
10314:
10309:
10306:
10303:
10298:
10294:
10288:
10283:
10280:
10277:
10274:
10269:
10265:
10261:
10258:
10255:
10239:
10238:
10234:
10233:
10230:absolute value
10200:
10199:
10198:
10185:
10181:
10174:
10168:
10162:
10156:
10150:
10147:
10141:
10136:
10133:
10130:
10125:
10121:
10117:
10114:
10111:
10095:
10094:
10090:
10089:
10083:
10076:
10075:
10074:
10061:
10057:
10050:
10044:
10038:
10032:
10026:
10023:
10019:
10015:
10012:
10004:
10000:
9996:
9993:
9987:
9983:
9978:
9972:
9968:
9961:
9957:
9954:
9949:
9946:
9931:
9930:
9926:
9925:
9924:
9923:
9909:
9905:
9898:
9892:
9886:
9880:
9877:
9872:
9866:
9861:
9845:
9844:
9840:
9839:
9838:
9837:
9823:
9819:
9812:
9806:
9800:
9797:
9792:
9786:
9781:
9765:
9764:
9760:
9759:
9750:
9749:
9748:
9735:
9731:
9724:
9718:
9712:
9709:
9704:
9701:
9696:
9690:
9687:
9682:
9677:
9671:
9666:
9660:
9655:
9651:
9644:
9629:
9628:
9624:
9623:
9622:
9621:
9608:
9604:
9597:
9591:
9585:
9582:
9577:
9574:
9562:
9548:
9544:
9537:
9531:
9525:
9522:
9517:
9511:
9506:
9490:
9489:
9485:
9484:
9478:
9471:
9464:
9457:
9450:
9449:
9448:
9435:
9431:
9424:
9418:
9415:
9410:
9407:
9402:
9395:
9391:
9386:
9381:
9377:
9370:
9355:
9354:
9350:
9349:
9348:
9347:
9334:
9330:
9323:
9317:
9314:
9308:
9304:
9289:
9288:
9284:
9283:
9282:
9281:
9268:
9264:
9257:
9251:
9248:
9242:
9238:
9223:
9222:
9218:
9217:
9206:
9205:
9204:
9191:
9187:
9180:
9177:
9171:
9166:
9161:
9157:
9140:
9139:
9135:
9134:
9133:
9132:
9119:
9115:
9108:
9105:
9097:
9092:
9089:
9086:
9079:
9074:
9071:
9068:
9062:
9057:
9054:
9039:
9038:
9034:
9033:
9032:
9031:
9018:
9014:
9007:
9004:
8998:
8994:
8979:
8978:
8974:
8973:
8958:
8957:
8956:
8943:
8939:
8932:
8929:
8923:
8918:
8915:
8909:
8903:
8893:
8889:
8883:
8880:
8875:
8872:
8869:
8865:
8860:
8855:
8851:
8847:
8842:
8838:
8821:
8820:
8816:
8815:
8812:
8811:
8810:
8797:
8793:
8786:
8783:
8777:
8774:
8769:
8764:
8757:
8752:
8746:
8741:
8738:
8733:
8727:
8721:
8716:
8710:
8705:
8701:
8694:
8682:
8669:
8665:
8658:
8655:
8650:
8647:
8635:
8622:
8618:
8611:
8608:
8603:
8596:
8592:
8584:
8580:
8573:
8565:
8562:
8552:
8539:
8535:
8528:
8525:
8516:
8512:
8504:
8500:
8495:
8491:
8479:
8466:
8462:
8455:
8452:
8447:
8441:
8436:
8431:
8413:
8412:
8408:
8407:
8406:
8405:
8392:
8388:
8381:
8378:
8374:
8362:
8357:
8353:
8347:
8344:
8338:
8334:
8328:
8325:
8319:
8315:
8309:
8306:
8302:
8296:
8291:
8285:
8282:
8279:
8276:
8273:
8268:
8265:
8262:
8259:
8256:
8250:
8238:
8224:
8220:
8213:
8210:
8202:
8197:
8193:
8181:
8168:
8164:
8157:
8154:
8149:
8143:
8138:
8133:
8118:
8105:
8101:
8094:
8091:
8086:
8083:
8068:
8067:
8063:
8062:
8061:
8060:
8047:
8043:
8036:
8033:
8019:
8014:
8010:
8004:
8001:
7995:
7991:
7985:
7982:
7976:
7972:
7966:
7963:
7959:
7954:
7950:
7944:
7941:
7936:
7931:
7928:
7923:
7918:
7915:
7910:
7905:
7902:
7897:
7894:
7890:
7886:
7876:
7863:
7859:
7852:
7849:
7844:
7838:
7833:
7828:
7820:
7815:
7810:
7805:
7793:
7778:
7774:
7767:
7764:
7759:
7754:
7746:
7742:
7737:
7734:
7729:
7724:
7718:
7715:
7711:
7696:
7695:
7691:
7690:
7689:
7688:
7675:
7671:
7667:
7661:
7657:
7645:
7632:
7628:
7624:
7617:
7613:
7607:
7603:
7587:
7586:
7582:
7581:
7580:
7579:
7566:
7562:
7558:
7552:
7549:
7543:
7538:
7535:
7520:
7519:
7511:
7510:
7509:
7508:
7495:
7491:
7487:
7478:
7473:
7470:
7465:
7462:
7447:
7446:
7442:
7441:
7440:
7439:
7426:
7422:
7415:
7412:
7407:
7403:
7397:
7394:
7384:
7371:
7367:
7360:
7357:
7352:
7347:
7342:
7330:
7317:
7313:
7306:
7303:
7298:
7293:
7288:
7276:
7263:
7259:
7255:
7249:
7245:
7233:
7218:
7194:
7174:
7169:
7165:
7161:
7158:
7155:
7152:
7149:
7146:
7133:
7132:
7128:
7127:
7126:
7125:
7112:
7108:
7104:
7101:
7098:
7093:
7088:
7083:
7056:omnicompetence
7041:
7040:
7039:
7026:
7022:
7018:
7013:
7008:
7003:
6988:
6987:
6983:
6982:
6981:
6980:
6967:
6963:
6959:
6954:
6951:
6936:
6935:
6910:
6907:
6900:
6896:
6892:
6887:
6880:
6876:
6872:
6867:
6860:
6856:
6852:
6847:
6840:
6836:
6832:
6827:
6822:
6819:
6814:
6811:
6808:
6793:
6792:
6779:
6775:
6768:
6765:
6758:
6754:
6750:
6745:
6738:
6734:
6730:
6725:
6720:
6717:
6712:
6709:
6694:
6685:Historically,
6682:
6679:
6665:
6664:
6663: (1896)).
6646:
6643:
6638:
6635:
6632:
6629:
6624:
6621:
6616:
6613:
6610:
6607:
6602:
6599:
6594:
6591:
6588:
6585:
6580:
6577:
6572:
6569:
6566:
6563:
6558:
6555:
6542:
6541:
6523:
6520:
6515:
6512:
6509:
6506:
6501:
6498:
6493:
6490:
6487:
6484:
6479:
6476:
6471:
6468:
6465:
6462:
6457:
6454:
6449:
6446:
6443:
6440:
6435:
6432:
6355:
6351:
6346:
6342:
6320:
6316:
6312:
6308:
6302:
6298:
6294:
6291:
6288:
6285:
6282:
6279:
6276:
6273:
6262:
6261:
6250:
6245:
6240:
6237:
6234:
6230:
6226:
6221:
6218:
6215:
6211:
6207:
6204:
6201:
6196:
6193:
6190:
6186:
6180:
6177:
6174:
6171:
6167:
6163:
6158:
6154:
6148:
6145:
6142:
6138:
6134:
6131:
6128:
6123:
6119:
6115:
6110:
6107:
6104:
6100:
6093:
6087:
6084:
6079:
6075:
6071:
6068:
6065:
6062:
6059:
6055:
6051:
6048:
6043:
6039:
6035:
6032:
6029:
6026:
6023:
6020:
6015:
6012:
6009:
6005:
5979:
5974:
5971:
5968:
5965:
5960:
5956:
5949:
5946:
5943:
5938:
5933:
5928:
5924:
5888:
5885:
5873:
5872:
5856:
5852:
5848:
5845:
5842:
5839:
5835:
5829:
5825:
5821:
5818:
5815:
5812:
5809:
5806:
5803:
5800:
5795:
5792:
5789:
5786:
5783:
5780:
5777:
5774:
5771:
5768:
5765:
5760:
5756:
5752:
5749:
5746:
5738:
5733:
5730:
5727:
5723:
5719:
5716:
5711:
5708:
5685:
5684:
5668:
5665:
5661:
5655:
5651:
5647:
5644:
5641:
5636:
5633:
5630:
5627:
5624:
5621:
5618:
5615:
5612:
5609:
5606:
5598:
5593:
5590:
5587:
5583:
5577:
5571:
5566:
5560:
5555:
5552:
5533:
5532:
5525:
5512:
5509:
5504:
5501:
5498:
5495:
5490:
5487:
5482:
5479:
5476:
5473:
5469:
5451:
5450:
5445:with the same
5431:
5415:
5410:
5405:
5401:
5398:
5395:
5391:
5385:
5382:
5377:
5374:
5370:
5364:
5361:
5356:
5353:
5349:
5343:
5340:
5335:
5332:
5329:
5321:
5318:
5315:
5312:
5309:
5306:
5303:
5289:
5285:
5281:
5276:
5272:
5258:
5253:
5250:
5247:
5243:
5239:
5233:
5230:
5227:
5224:
5221:
5218:
5215:
5212:
5207:
5204:
5196:
5191:
5188:
5185:
5181:
5177:
5174:
5172:
5168:
5165:
5160:
5159:
5156:
5153:
5145:
5141:
5135:
5131:
5127:
5124:
5121:
5115:
5111:
5102:
5099:
5096:
5091:
5088:
5085:
5079:
5071:
5067:
5061:
5057:
5053:
5050:
5047:
5041:
5037:
5029:
5026:
5021:
5013:
5009:
5005:
5002:
4998:
4993:
4985:
4981:
4975:
4971:
4967:
4964:
4961:
4957:
4954:
4951:
4948:
4945:
4942:
4939:
4936:
4929:
4926:
4922:
4917:
4914:
4911:
4908:
4905:
4902:
4894:
4889:
4886:
4883:
4879:
4870:
4866:
4862:
4859:
4855:
4850:
4847:
4845:
4843:
4840:
4837:
4834:
4833:
4816:
4815:
4803:
4799:
4796:
4788:
4784:
4780:
4777:
4773:
4768:
4760:
4756:
4752:
4749:
4745:
4740:
4732:
4728:
4724:
4721:
4717:
4712:
4704:
4700:
4696:
4693:
4689:
4683:
4677:
4672:
4666:
4663:
4660:
4657:
4650:
4646:
4640:
4637:
4632:
4629:
4621:
4616:
4613:
4610:
4606:
4600:
4595:
4589:
4586:
4583:
4580:
4573:
4570:
4566:
4562:
4559:
4556:
4548:
4543:
4540:
4537:
4533:
4527:
4522:
4519:
4502:
4501:
4486:
4483:
4480:
4478:
4476:
4455:
4452:
4449:
4444:
4441:
4436:
4433:
4428:
4425:
4420:
4417:
4412:
4409:
4404:
4401:
4396:
4393:
4388:
4385:
4380:
4377:
4374:
4372:
4370:
4367:
4366:
4337:
4334:
4327:
4326:
4313:
4310:
4305:
4302:
4299:
4296:
4291:
4288:
4283:
4280:
4277:
4274:
4269:
4266:
4261:
4258:
4255:
4252:
4247:
4244:
4239:
4236:
4233:
4230:
4225:
4222:
4206:
4205:
4192:
4189:
4184:
4181:
4178:
4175:
4170:
4167:
4162:
4159:
4156:
4153:
4148:
4145:
4140:
4137:
4134:
4131:
4126:
4123:
4118:
4115:
4112:
4109:
4104:
4101:
4082:
4081:
4068:
4065:
4060:
4057:
4054:
4051:
4046:
4043:
4038:
4035:
4032:
4029:
4024:
4021:
4016:
4013:
4010:
4007:
4002:
3999:
3983:
3982:
3969:
3966:
3961:
3958:
3955:
3950:
3947:
3942:
3939:
3936:
3933:
3928:
3925:
3920:
3917:
3914:
3911:
3906:
3903:
3881:For instance,
3861: β 2
3833:
3830:
3827:
3824:
3821:
3818:
3813:
3809:
3805:
3800:
3796:
3792:
3789:
3786:
3783:
3780:
3777:
3774:
3769:
3765:
3761:
3758:
3755:
3752:
3719:
3718:
3705:
3702:
3697:
3694:
3691:
3686:
3683:
3678:
3675:
3672:
3669:
3664:
3661:
3637:Main article:
3634:
3631:
3616:François Viète
3573:
3572:
3561:
3556:
3553:
3548:
3545:
3542:
3539:
3534:
3531:
3526:
3510:
3501:
3494:
3487:
3480:
3473:
3466:
3459:
3452:
3451:
3440:
3433:
3430:
3426:
3420:
3416:
3410:
3405:
3402:
3398:
3392:
3384:
3380:
3376:
3371:
3367:
3359:
3355:
3349:
3345:
3341:
3335:
3330:
3327:
3323:
3298:
3287:
3270:
3267:
3262:Main article:
3259:
3256:
3235:
3234:
3233:is irrational.
3220:
3211:," thick (cf.
3185:Rabbi Nehemiah
3145:implies that "
3132:
3129:
3085:Main article:
3082:
3079:
3048:
3042:
3004:relative error
2985:
2982:
2981:
2980:
2973:
2966:
2955:
2940:
2937:
2922:
2911:
2904:
2897:
2894:
2883:
2872:
2859:
2858:Recent records
2856:
2844:Hitachi SR8000
2812:
2811:
2800:
2792:
2788:
2784:
2781:
2778:
2775:
2771:
2765:
2761:
2757:
2754:
2751:
2748:
2745:
2742:
2739:
2736:
2731:
2728:
2725:
2722:
2719:
2716:
2713:
2710:
2707:
2704:
2701:
2696:
2692:
2688:
2685:
2682:
2674:
2669:
2666:
2663:
2659:
2655:
2652:
2647:
2644:
2572:William Shanks
2542:
2541:
2530:
2527:
2519:
2514:
2510:
2505:
2502:
2480:
2479:
2463:
2460:
2456:
2450:
2446:
2442:
2439:
2436:
2431:
2428:
2425:
2422:
2419:
2416:
2413:
2410:
2407:
2404:
2401:
2393:
2388:
2385:
2382:
2378:
2372:
2366:
2361:
2355:
2350:
2347:
2319:Main article:
2316:
2313:
2300:William Shanks
2229:
2226:
2223:
2220:
2217:
2214:
2211:
2208:
2205:
2202:
2196:
2193:
2149:
2145:
2140:
2133:
2130:
2125:
2120:
2117:
2110:
2105:
2100:
2093:
2090:
2085:
2080:
2077:
2070:
2065:
2060:
2053:
2050:
2045:
2040:
2037:
2030:
2025:
2020:
2013:
2010:
2005:
2000:
1997:
1990:
1985:
1981:
1974:
1971:
1968:
1965:
1960:
1957:
1951:
1945:
1942:
1939:
1936:
1931:
1928:
1921:
1915:
1910:
1907:
1904:
1900:
1896:
1890:
1887:
1882:
1878:
1874:
1867:
1863:
1859:
1851:
1846:
1843:
1840:
1836:
1832:
1827:
1824:
1809:Wallis product
1807:published the
1748:François Viète
1743:
1740:
1732:
1731:
1720:
1717:
1714:
1700:
1699:
1688:
1685:
1682:
1679:
1676:
1632:
1631:
1617:
1614:
1611:
1606:
1602:
1598:
1593:
1590:
1585:
1581:
1574:
1568:
1565:
1562:
1559:
1552:
1548:
1544:
1541:
1538:
1532:
1529:
1526:
1521:
1518:
1513:
1508:
1505:
1500:
1495:
1492:
1487:
1484:
1481:
1478:
1474:
1470:
1423:
1403:
1402:
1390:
1386:
1383:
1375:
1371:
1367:
1364:
1360:
1355:
1347:
1343:
1339:
1336:
1332:
1327:
1321:
1318:
1315:
1311:
1306:
1303:
1299:
1293:
1288:
1282:
1279:
1276:
1273:
1266:
1262:
1256:
1253:
1248:
1245:
1237:
1232:
1229:
1226:
1222:
1216:
1211:
1205:
1202:
1199:
1196:
1189:
1186:
1182:
1178:
1175:
1172:
1164:
1159:
1156:
1153:
1149:
1143:
1138:
1135:
1113:
1110:
1105:
1099:
1095:
1092:
1089:
1086:
1083:
1080:
1077:
1066:
1065:
1053:
1049:
1046:
1041:
1038:
1033:
1028:
1025:
1020:
1015:
1012:
1007:
1004:
1000:
996:
993:
990:
968:
965:
962:
959:
956:
953:
950:
947:
944:
908:
905:
869:Approximating
857:
736:
731:
727:
722:
718:
715:
712:
708:
704:
701:
698:
622:
551:Middle Kingdom
545:(dated to the
462:
459:
436:William Shanks
383:Approximations
378:
377:
375:
374:
367:
360:
352:
349:
348:
347:
346:
338:
330:
325:
317:
316:
315:Related topics
312:
311:
310:
309:
304:
296:
295:
291:
290:
289:
288:
281:
273:
272:
268:
267:
266:
265:
260:
255:
250:
245:
243:William Shanks
240:
235:
230:
225:
220:
218:François Viète
215:
210:
205:
200:
195:
190:
185:
177:
176:
172:
171:
170:
169:
164:
159:
157:Approximations
154:
152:Less than 22/7
146:
145:
141:
140:
139:
138:
133:
125:
124:
120:
119:
118:
117:
112:
107:
99:
98:
94:
93:
73:
72:
63:
62:
15:
9:
6:
4:
3:
2:
22392:
22381:
22378:
22376:
22375:Pi algorithms
22373:
22371:
22368:
22366:
22363:
22361:
22358:
22357:
22355:
22344:
22338:
22334:
22329:
22325:
22323:9780563488033
22319:
22315:
22310:
22306:
22300:
22296:
22291:
22287:
22281:
22277:
22272:
22268:
22264:
22260:
22254:
22250:
22249:
22245:A History of
22241:
22237:
22232:
22227:
22223:
22219:
22215:
22211:
22204:
22200:
22196:
22192:
22188:
22187:
22166:
22162:
22155:
22148:
22142:
22127:
22120:
22104:
22098:
22082:
22078:
22074:
22068:
22062:
22056:
22047:
22042:
22035:
22026:
22025:
22020:
22019:"BBP Formula"
22017:
22010:
22003:
21999:
21995:
21991:
21987:
21983:
21976:
21968:
21964:
21960:
21956:
21952:
21948:
21944:
21940:
21936:
21929:
21921:
21915:
21907:
21906:
21900:
21896:
21890:
21882:
21881:
21875:
21871:
21865:
21857:
21851:
21847:
21840:
21833:
21829:
21823:
21815:
21809:
21805:
21798:
21796:
21794:
21792:
21790:
21781:
21775:
21771:
21764:
21757:(5): 456β458.
21756:
21752:
21741:
21732:
21727:
21723:
21719:
21715:
21711:
21704:
21696:
21692:
21688:
21684:
21680:
21676:
21669:
21667:
21658:
21654:
21650:
21646:
21642:
21639:
21638:
21633:
21626:
21620:
21616:
21613:
21605:
21597:
21591:
21587:
21583:
21577:
21569:
21565:
21558:
21550:
21549:
21541:
21533:
21532:
21524:
21516:
21512:
21508:
21502:
21498:
21494:
21489:
21484:
21480:
21473:
21471:
21469:
21462:
21460:
21456:
21452:
21448:
21444:
21440:
21434:
21431:
21427:
21423:
21419:
21415:
21410:
21407:
21403:
21402:
21397:
21393:
21388:
21385:
21381:
21377:
21376:Newton, Isaac
21372:
21369:
21365:Reprinted in
21362:
21355:
21349:
21346:
21337:
21330:
21329:Beckmann 1971
21325:
21319:, p. 119
21318:
21313:
21311:
21304:, p. 118
21303:
21298:
21291:
21290:Beckmann 1971
21286:
21280:, p. 131
21279:
21274:
21268:
21263:
21247:
21243:
21236:
21220:
21216:
21209:
21202:
21201:
21196:
21191:
21176:
21172:
21167:
21162:
21158:
21154:
21147:
21136:
21128:
21124:
21120:
21116:
21109:
21090:
21086:
21079:
21073:
21058:
21054:
21047:
21032:
21028:
21021:
21006:
21002:
20995:
20980:
20976:
20969:
20954:. 8 June 2022
20953:
20952:
20947:
20941:
20925:
20924:
20919:
20913:
20898:
20892:
20876:
20870:
20855:
20849:
20834:
20830:
20824:
20815:
20810:
20803:
20788:
20784:
20777:
20775:
20773:
20771:
20762:
20755:
20753:
20744:
20737:
20722:
20721:New Scientist
20718:
20711:
20703:
20697:
20691:
20687:
20684:
20679:
20671:
20665:
20649:
20645:
20641:
20635:
20627:
20623:
20619:
20615:
20612:(77): 76β99.
20611:
20607:
20599:
20595:
20589:
20587:
20585:
20583:
20581:
20574:
20570:
20564:
20548:
20547:
20542:
20536:
20528:
20524:
20520:
20516:
20512:
20508:
20504:
20500:
20493:
20485:
20481:
20477:
20476:
20468:
20460:
20456:
20452:
20448:
20443:
20438:
20434:
20430:
20427:(3985): 342.
20426:
20422:
20421:
20416:
20409:
20407:
20398:
20397:
20392:
20386:
20380:
20378:
20371:
20367:
20363:
20359:
20355:
20351:
20345:
20341:
20337:
20330:
20322:
20318:
20314:
20299:
20291:
20287:
20283:
20279:
20275:
20271:
20267:
20263:
20252:
20246:
20243:
20239:
20230:Reprinted in
20227:
20222:
20221:
20216:
20215:
20210:
20205:
20201:
20183:
20180:
20177:
20170:
20167:
20153:
20149:
20145:
20139:
20131:
20127:
20123:
20111:
20108:
20102:
20088:
20084:
20080:
20074:
20066:
20062:
20058:
20046:
20043:
20037:
20025:
20022:
20016:
20010:
20007:
19992:
19988:
19984:
19980:
19976:
19970:
19966:
19962:
19961:
19956:
19950:
19943:(2): 207β212.
19942:
19938:
19931:
19924:
19906:
19899:
19890:
19885:
19881:
19877:
19873:
19866:
19858:
19854:
19850:
19843:
19835:
19828:
19821:(1β4): 68β71.
19820:
19816:
19809:
19801:
19797:
19793:
19786:
19784:
19775:
19769:
19765:
19761:
19754:
19746:
19740:
19736:
19729:
19713:
19709:
19703:
19696:. p. 70.
19695:
19691:
19672:
19668:
19664:
19663:
19660:
19656:ΔryabhaαΉΔ«ya (
19653:
19647:
19643:
19640:
19635:
19626:
19620:
19616:
19615:
19607:
19603:
19598:
19593:
19589:
19585:
19578:
19570:
19566:
19562:
19555:
19546:
19541:
19534:
19526:
19522:
19518:
19514:
19509:
19504:
19500:
19496:
19492:
19485:
19478:
19472:
19464:
19458:
19454:
19450:
19446:
19440:
19433:
19432:Beckmann 1971
19427:
19419:
19412:
19401:
19394:
19387:
19379:
19373:
19369:
19365:
19364:
19359:
19353:
19346:
19342:
19334:
19333:Beckmann 1971
19330:
19324:
19308:
19304:
19303:
19295:
19287:
19281:
19277:
19273:
19266:
19264:
19256:
19254:
19250:
19247:Based on the
19244:
19238:
19234:
19230:
19226:
19220:
19212:
19205:
19190:
19186:
19179:
19171:
19167:
19163:
19162:
19157:
19150:
19148:
19146:
19144:
19139:
19129:
19126:
19124:
19121:
19119:
19116:
19114:
19111:
19110:
19101:
19096:
19095:
19091:
19088:
19087:Stu's Pi page
19083:
19063:
19057:
19054:
19051:
19050:Stu's Pi page
19047:
19042:
19031:
19030:
19024:
19020:
19016:
19013:
19009:
19004:
19000:
18999:
18996:
18990:
18987:
18984:
18983:
18982:
18980:
18979:disk swapping
18976:
18975:checkpointing
18962:
18960:
18956:
18952:
18948:
18944:
18935:
18933:
18925:
18915:
18913:
18909:
18892:
18885:
18866:
18863:
18861:
18858:
18857:
18837:
18829:
18826:
18823:
18814:
18808:
18802:
18795:
18793:
18791:
18787:
18784:
18783:
18763:
18755:
18749:
18746:
18743:
18737:
18730:
18727:
18725:
18722:
18721:
18700:
18694:
18691:
18685:
18679:
18673:
18666:
18663:
18661:
18658:
18657:
18653:
18650:
18647:
18646:
18643:
18620:
18617:
18609:
18606:
18598:
18590:
18587:
18581:
18575:
18572:
18561:
18558:
18555:
18552:
18546:
18540:
18537:
18521:
18518:
18515:
18511:
18502:
18497:
18493:
18488:
18483:
18480:
18471:
18470:
18469:
18467:
18463:
18459:
18454:
18444:
18441:
18422:
18419:
18415:
18409:
18401:
18398:
18387:
18384:
18381:
18378:
18372:
18366:
18363:
18347:
18344:
18341:
18337:
18331:
18325:
18320:
18314:
18309:
18306:
18297:
18295:
18292:
18289:
18274:
18269:
18264:
18260:
18257:
18254:
18250:
18244:
18241:
18236:
18233:
18229:
18223:
18220:
18215:
18212:
18208:
18202:
18199:
18194:
18191:
18188:
18182:
18176:
18173:
18170:
18167:
18157:
18153:
18149:
18144:
18140:
18126:
18123:
18120:
18116:
18112:
18106:
18103:
18097:
18094:
18091:
18088:
18080:
18077:
18064:
18061:
18058:
18054:
18050:
18045:
18042:
18033:
18032:
18031:
18009:
18002:
17999:
17996:
17993:
17989:
17984:
17978:
17975:
17972:
17969:
17963:
17959:
17953:
17947:
17944:
17941:
17938:
17932:
17928:
17922:
17916:
17913:
17910:
17907:
17901:
17897:
17891:
17885:
17882:
17879:
17876:
17870:
17866:
17860:
17854:
17851:
17848:
17845:
17841:
17836:
17830:
17827:
17824:
17821:
17815:
17811:
17805:
17801:
17793:
17790:
17786:
17780:
17772:
17769:
17752:
17749:
17746:
17742:
17734:
17730:
17726:
17721:
17718:
17711:
17710:
17709:
17707:
17703:
17699:
17693:
17692:quadrillionth
17689:
17685:
17681:
17673:
17669:
17665:
17646:
17642:
17635:
17632:
17629:
17626:
17622:
17617:
17611:
17608:
17605:
17602:
17598:
17593:
17587:
17584:
17581:
17578:
17574:
17569:
17563:
17560:
17557:
17554:
17550:
17544:
17536:
17532:
17528:
17516:
17513:
17510:
17506:
17502:
17499:
17492:
17491:
17490:
17488:
17480:
17479:a new formula
17476:
17475:Simon Plouffe
17472:
17471:Peter Borwein
17468:
17463:
17461:
17457:
17453:
17444:
17442:
17438:
17409:
17402:
17399:
17396:
17393:
17389:
17384:
17378:
17375:
17372:
17369:
17363:
17359:
17353:
17347:
17344:
17341:
17338:
17332:
17328:
17322:
17316:
17313:
17310:
17307:
17301:
17297:
17291:
17285:
17282:
17279:
17276:
17270:
17266:
17260:
17254:
17251:
17248:
17245:
17241:
17236:
17230:
17227:
17224:
17221:
17215:
17211:
17205:
17201:
17193:
17190:
17186:
17179:
17171:
17168:
17152:
17149:
17146:
17142:
17134:
17130:
17126:
17121:
17118:
17111:
17110:
17109:
17099:
17093:
17089:
17066:
17060:
17057:
17047:
17043:
17039:
17034:
17030:
17026:
17013:
17010:
17007:
17003:
16999:
16996:
16993:
16990:
16983:
16982:
16981:
16973:
16969:
16965:
16927:
16922:
16917:
16914:
16909:
16903:
16896:
16893:
16890:
16887:
16883:
16878:
16872:
16869:
16866:
16863:
16859:
16854:
16848:
16845:
16842:
16839:
16835:
16830:
16824:
16821:
16818:
16815:
16811:
16805:
16794:
16791:
16788:
16784:
16780:
16777:
16770:
16769:
16768:
16762:
16752:
16742:
16721:
16718:
16713:
16710:
16705:
16700:
16697:
16692:
16687:
16684:
16679:
16674:
16671:
16666:
16661:
16658:
16653:
16648:
16645:
16640:
16635:
16632:
16627:
16624:
16621:
16619:
16605:
16602:
16599:
16596:
16588:
16584:
16570:
16566:
16563:
16551:
16548:
16535:
16532:
16529:
16525:
16521:
16517:
16514:
16508:
16505:
16500:
16496:
16491:
16486:
16480:
16477:
16472:
16468:
16463:
16458:
16452:
16449:
16444:
16440:
16435:
16430:
16424:
16421:
16416:
16412:
16407:
16402:
16400:
16391:
16386:
16383:
16375:
16371:
16367:
16364:
16361:
16358:
16355:
16352:
16349:
16346:
16341:
16338:
16335:
16332:
16329:
16323:
16315:
16311:
16307:
16304:
16301:
16298:
16295:
16292:
16287:
16284:
16281:
16275:
16267:
16263:
16259:
16256:
16253:
16250:
16246:
16241:
16236:
16233:
16227:
16223:
16220:
16216:
16211:
16208:
16203:
16199:
16194:
16191:
16187:
16183:
16180:
16178:
16173:
16162:
16161:
16160:
16141:
16138:
16133:
16129:
16124:
16121:
16116:
16112:
16109:
16102:
16101:
16100:
16092:
16078:
16070:
16066:
16048:
16044:
16020:
16013:
16009:
16002:
15999:
15996:
15992:
15988:
15985:
15979:
15976:
15971:
15968:
15965:
15961:
15957:
15952:
15949:
15940:
15939:
15938:
15921:
15918:
15913:
15910:
15905:
15902:
15899:
15894:
15891:
15886:
15883:
15880:
15875:
15872:
15867:
15864:
15861:
15856:
15853:
15848:
15845:
15842:
15835:
15832:
15829:
15826:
15822:
15818:
15813:
15810:
15800:
15797:
15794:
15790:
15786:
15781:
15778:
15769:
15768:
15767:
15753:
15746:The constant
15744:
15730:
15710:
15690:
15663:
15660:
15656:
15649:
15641:
15638:
15635:
15631:
15627:
15624:
15620:
15614:
15610:
15605:
15601:
15598:
15595:
15591:
15583:
15575:
15572:
15569:
15565:
15561:
15555:
15547:
15543:
15534:
15531:
15527:
15520:
15512:
15509:
15506:
15502:
15498:
15495:
15491:
15485:
15481:
15476:
15472:
15469:
15466:
15462:
15454:
15446:
15443:
15440:
15436:
15432:
15426:
15418:
15414:
15405:
15402:
15399:
15393:
15385:
15381:
15372:
15369:
15365:
15361:
15358:
15352:
15344:
15340:
15327:
15326:
15325:
15308:
15300:
15297:
15294:
15288:
15283:
15279:
15275:
15271:
15268:
15265:
15259:
15254:
15250:
15243:
15240:
15237:
15230:
15225:
15213:
15210:
15207:
15204:
15200:
15187:
15184:
15181:
15177:
15173:
15170:
15164:
15158:
15155:
15148:
15147:
15146:
15129:
15121:
15118:
15115:
15105:
15101:
15097:
15094:
15085:
15082:
15079:
15076:
15072:
15062:
15056:
15053:
15050:
15047:
15037:
15029:
15026:
15018:
15015:
15011:
14997:
14994:
14991:
14987:
14983:
14977:
14971:
14968:
14961:
14960:
14959:
14957:
14939:
14934:
14929:
14925:
14900:
14897:
14894:
14890:
14886:
14883:
14878:
14873:
14869:
14845:
14842:
14839:
14836:
14831:
14824:
14820:
14813:
14810:
14807:
14803:
14799:
14796:
14790:
14787:
14784:
14777:
14774:
14771:
14767:
14763:
14754:
14753:
14752:
14738:
14727:
14724:
14720:
14705:
14700:
14695:
14691:
14688:
14685:
14681:
14675:
14672:
14667:
14664:
14660:
14654:
14651:
14646:
14643:
14639:
14633:
14630:
14625:
14622:
14619:
14616:
14609:
14608:
14607:
14586:
14583:
14578:
14575:
14570:
14565:
14562:
14557:
14552:
14549:
14544:
14539:
14536:
14531:
14526:
14523:
14518:
14513:
14510:
14505:
14500:
14497:
14492:
14487:
14484:
14479:
14474:
14471:
14466:
14463:
14460:
14458:
14442:
14439:
14436:
14433:
14419:
14415:
14412:
14389:
14386:
14383:
14379:
14360:
14357:
14354:
14350:
14346:
14338:
14332:
14329:
14326:
14323:
14306:
14302:
14298:
14293:
14290:
14287:
14283:
14264:
14261:
14258:
14254:
14250:
14242:
14239:
14233:
14230:
14227:
14224:
14209:
14206:
14188:
14185:
14182:
14178:
14174:
14171:
14169:
14160:
14156:
14153:
14145:
14142:
14139:
14136:
14133:
14130:
14127:
14124:
14121:
14109:
14106:
14103:
14100:
14097:
14094:
14091:
14088:
14085:
14074:
14066:
14063:
14060:
14057:
14054:
14051:
14048:
14036:
14033:
14030:
14027:
14024:
14021:
14018:
14007:
13999:
13996:
13993:
13990:
13987:
13975:
13972:
13969:
13966:
13963:
13952:
13944:
13941:
13938:
13926:
13923:
13920:
13909:
13901:
13889:
13878:
13875:
13871:
13867:
13864:
13862:
13857:
13846:
13845:
13844:
13836:
13826:
13824:
13808:
13788:
13779:
13774:
13754:
13750:
13747:
13744:
13739:
13736:
13731:
13726:
13723:
13718:
13713:
13710:
13705:
13700:
13697:
13691:
13687:
13684:
13676:
13673:
13670:
13667:
13653:
13645:
13642:
13621:
13618:
13615:
13611:
13607:
13604:
13601:
13594:
13593:
13592:
13591:
13576:
13574:
13531:
13528:
13525:
13520:
13516:
13512:
13507:
13504:
13499:
13495:
13488:
13485:
13473:
13470:
13464:
13460:
13454:
13451:
13448:
13442:
13438:
13432:
13429:
13426:
13422:
13413:
13412:
13411:
13391:
13388:
13385:
13382:
13379:
13376:
13373:
13370:
13367:
13340:
13337:
13334:
13331:
13317:
13313:
13301:
13298:
13295:
13281:
13277:
13265:
13262:
13259:
13245:
13241:
13229:
13226:
13223:
13209:
13201:
13198:
13192:
13181:
13175:
13172:
13169:
13166:
13159:
13156:
13153:
13145:
13142:
13131:
13126:
13123:
13120:
13116:
13112:
13109:
13106:
13099:
13098:
13097:
13095:
13051:
13048:
13045:
13031:
13027:
13015:
13012:
12998:
12994:
12982:
12979:
12965:
12961:
12949:
12946:
12932:
12928:
12916:
12913:
12899:
12895:
12883:
12880:
12877:
12849:
12846:
12843:
12829:
12825:
12813:
12810:
12796:
12792:
12780:
12777:
12763:
12759:
12747:
12744:
12730:
12726:
12714:
12711:
12699:
12688:
12685:
12678:
12647:
12644:
12641:
12627:
12623:
12611:
12608:
12594:
12590:
12578:
12575:
12561:
12557:
12545:
12542:
12538:
12535:
12528:
12527:
12526:
12524:
12516:
12506:
12499:
12497:
12487:
12484:
12481:
12480:
12476:
12473:
12470:
12469:
12465:
12462:
12459:
12458:
12454:
12451:
12448:
12447:
12443:
12440:
12437:
12436:
12432:
12429:
12426:
12425:
12421:
12418:
12415:
12414:
12410:
12407:
12404:
12403:
12396:
12393:
12390:
12389:
12386:
12366:
12362:
12354:
12349:
12338:
12335:
12332:
12329:
12326:
12323:
12320:
12317:
12314:
12311:
12308:
12307:
12303:
12300:
12297:
12294:
12291:
12288:
12285:
12282:
12279:
12276:
12273:
12272:
12268:
12265:
12262:
12259:
12256:
12253:
12250:
12247:
12244:
12241:
12238:
12237:
12233:
12230:
12227:
12224:
12221:
12218:
12215:
12212:
12209:
12206:
12203:
12202:
12198:
12195:
12192:
12189:
12186:
12183:
12180:
12177:
12174:
12171:
12168:
12167:
12163:
12160:
12157:
12154:
12151:
12148:
12145:
12142:
12139:
12136:
12133:
12132:
12128:
12125:
12122:
12119:
12116:
12113:
12110:
12107:
12104:
12101:
12098:
12097:
12093:
12090:
12087:
12084:
12081:
12078:
12075:
12072:
12069:
12066:
12063:
12062:
12058:
12055:
12052:
12049:
12046:
12043:
12040:
12037:
12034:
12031:
12028:
12027:
12023:
12020:
12017:
12014:
12011:
12008:
12005:
12002:
11999:
11996:
11993:
11992:
11988:
11985:
11982:
11979:
11976:
11973:
11970:
11967:
11964:
11961:
11958:
11957:
11954:
11953:
11952:
11934:
11919:
11913:
11907:
11901:
11870:
11867:
11864:
11857:
11853:
11849:
11844:
11840:
11827:
11820:
11817:
11810:
11806:
11802:
11797:
11793:
11780:
11774:
11767:
11762:
11759:
11756:
11753:
11749:
11742:
11737:
11734:
11731:
11728:
11724:
11716:
11712:
11708:
11695:
11687:
11684:
11677:
11676:
11675:
11672:
11647:
11644:
11641:
11634:
11630:
11626:
11621:
11617:
11607:
11606:
11605:
11602:
11596:
11583:
11578:
11572:
11566:
11560:
11540:
11533:
11529:
11525:
11520:
11516:
11510:
11507:
11500:
11499:
11498:
11495:
11489:
11484:
11480:
11474:
11453:
11448:
11444:
11440:
11437:
11434:
11427:
11426:
11425:
11410:
11380:
11377:
11367:
11349:
11346:
11341:
11336:
11333:
11328:
11323:
11320:
11315:
11310:
11307:
11302:
11297:
11294:
11289:
11284:
11281:
11276:
11271:
11268:
11263:
11258:
11255:
11250:
11245:
11242:
11237:
11232:
11229:
11224:
11219:
11216:
11211:
11206:
11203:
11194:
11193:
11192:
11191:
11183:
11175:
11171:
11170:
11150:
11144:
11139:
11133:
11130:
11127:
11120:
11106:
11103:
11100:
11095:
11092:
11083:
11082:
11081:
11080:
11076:
11075:
11071:
11067:
11063:
11045:
11042:
11037:
11034:
11026:
11023:, namely the
11022:
11018:
11017:Daniel Shanks
11014:
10989:
10984:
10981:
10978:
10969:
10966:
10960:
10958:
10953:
10941:
10936:
10933:
10930:
10924:
10922:
10917:
10905:
10900:
10897:
10892:
10887:
10878:
10875:
10869:
10867:
10862:
10850:
10845:
10842:
10839:
10830:
10827:
10821:
10819:
10814:
10803:
10802:
10800:
10779:
10776:
10771:
10767:
10761:
10758:
10747:
10744:
10739:
10735:
10729:
10726:
10718:
10708:
10705:
10700:
10696:
10690:
10687:
10679:
10669:
10666:
10661:
10657:
10651:
10648:
10642:
10639:
10632:
10631:
10629:
10625:
10606:
10594:
10591:
10586:
10578:
10575:
10562:
10559:
10549:
10548:
10547:
10546:
10542:
10541:
10533:
10528:
10507:
10500:
10493:
10483:
10480:
10475:
10464:
10461:
10456:
10443:
10438:
10433:
10420:
10415:
10410:
10397:
10392:
10389:
10378:
10375:
10371:
10364:
10361:
10351:
10350:
10349:
10348:
10344:
10343:
10339:
10335:
10331:
10312:
10304:
10301:
10296:
10286:
10281:
10278:
10275:
10267:
10263:
10256:
10253:
10243:
10242:
10241:
10240:
10236:
10235:
10231:
10227:
10223:
10220:
10216:
10213:
10212:discriminants
10210:and negative
10209:
10205:
10201:
10183:
10179:
10172:
10166:
10160:
10154:
10148:
10145:
10139:
10131:
10128:
10123:
10119:
10112:
10109:
10099:
10098:
10097:
10096:
10092:
10091:
10082:
10077:
10059:
10055:
10048:
10042:
10036:
10030:
10024:
10021:
10017:
10013:
10010:
10002:
9994:
9991:
9985:
9981:
9970:
9966:
9959:
9955:
9952:
9947:
9944:
9935:
9934:
9933:
9932:
9928:
9927:
9907:
9903:
9896:
9890:
9884:
9878:
9875:
9870:
9864:
9859:
9849:
9848:
9847:
9846:
9842:
9841:
9821:
9817:
9810:
9804:
9798:
9795:
9790:
9784:
9779:
9769:
9768:
9767:
9766:
9762:
9761:
9756:
9751:
9733:
9729:
9722:
9716:
9710:
9707:
9702:
9699:
9694:
9688:
9685:
9680:
9675:
9669:
9664:
9658:
9653:
9649:
9642:
9633:
9632:
9631:
9630:
9626:
9625:
9606:
9602:
9595:
9589:
9583:
9580:
9575:
9572:
9563:
9546:
9542:
9535:
9529:
9523:
9520:
9516:1130173253125
9515:
9509:
9505:2510613731736
9504:
9494:
9493:
9492:
9491:
9487:
9486:
9477:
9470:
9463:
9456:
9451:
9433:
9429:
9422:
9416:
9413:
9408:
9405:
9400:
9393:
9389:
9384:
9379:
9375:
9368:
9359:
9358:
9357:
9356:
9352:
9351:
9332:
9328:
9321:
9315:
9312:
9306:
9302:
9293:
9292:
9291:
9290:
9286:
9285:
9266:
9262:
9255:
9249:
9246:
9240:
9236:
9227:
9226:
9225:
9224:
9220:
9219:
9207:
9189:
9185:
9178:
9175:
9169:
9164:
9159:
9155:
9144:
9143:
9142:
9141:
9137:
9136:
9117:
9113:
9106:
9103:
9095:
9090:
9087:
9084:
9077:
9072:
9069:
9066:
9060:
9055:
9052:
9043:
9042:
9041:
9040:
9036:
9035:
9016:
9012:
9005:
9002:
8996:
8992:
8983:
8982:
8981:
8980:
8976:
8975:
8967:
8963:
8960:This is from
8959:
8941:
8937:
8930:
8927:
8921:
8916:
8913:
8907:
8901:
8891:
8881:
8878:
8870:
8867:
8863:
8858:
8853:
8849:
8845:
8840:
8836:
8825:
8824:
8823:
8822:
8818:
8817:
8813:
8795:
8791:
8784:
8781:
8775:
8772:
8767:
8762:
8755:
8750:
8744:
8739:
8736:
8731:
8725:
8719:
8714:
8708:
8703:
8699:
8692:
8683:
8667:
8663:
8656:
8653:
8648:
8645:
8636:
8620:
8616:
8609:
8606:
8601:
8594:
8590:
8582:
8578:
8571:
8563:
8560:
8553:
8537:
8533:
8526:
8523:
8514:
8510:
8502:
8498:
8493:
8489:
8480:
8464:
8460:
8453:
8450:
8445:
8439:
8434:
8429:
8417:
8416:
8415:
8414:
8410:
8409:
8390:
8386:
8379:
8376:
8372:
8360:
8355:
8351:
8345:
8342:
8336:
8332:
8326:
8323:
8317:
8313:
8307:
8304:
8300:
8294:
8289:
8283:
8280:
8277:
8274:
8271:
8266:
8263:
8260:
8257:
8254:
8248:
8239:
8222:
8218:
8211:
8208:
8200:
8195:
8191:
8182:
8166:
8162:
8155:
8152:
8147:
8141:
8136:
8131:
8119:
8103:
8099:
8092:
8089:
8084:
8081:
8072:
8071:
8070:
8069:
8065:
8064:
8045:
8041:
8034:
8031:
8017:
8012:
8008:
8002:
7999:
7993:
7989:
7983:
7980:
7974:
7970:
7964:
7961:
7957:
7952:
7948:
7942:
7939:
7934:
7929:
7926:
7921:
7916:
7913:
7908:
7903:
7900:
7895:
7892:
7888:
7884:
7877:
7861:
7857:
7850:
7847:
7842:
7836:
7831:
7826:
7818:
7813:
7808:
7803:
7794:
7792:
7776:
7772:
7765:
7762:
7757:
7752:
7744:
7740:
7735:
7732:
7727:
7722:
7716:
7713:
7709:
7700:
7699:
7698:
7697:
7693:
7692:
7673:
7669:
7665:
7659:
7655:
7646:
7630:
7626:
7622:
7615:
7611:
7605:
7601:
7591:
7590:
7589:
7588:
7584:
7583:
7564:
7560:
7556:
7550:
7547:
7541:
7536:
7533:
7524:
7523:
7522:
7521:
7517:
7513:
7512:
7493:
7489:
7485:
7476:
7471:
7468:
7463:
7460:
7451:
7450:
7449:
7448:
7444:
7443:
7424:
7420:
7413:
7410:
7405:
7401:
7395:
7392:
7385:
7369:
7365:
7358:
7355:
7350:
7345:
7340:
7331:
7315:
7311:
7304:
7301:
7296:
7291:
7286:
7277:
7261:
7257:
7253:
7247:
7243:
7234:
7232:
7216:
7208:
7192:
7172:
7167:
7163:
7159:
7156:
7153:
7150:
7147:
7144:
7137:
7136:
7135:
7134:
7130:
7129:
7110:
7106:
7102:
7099:
7096:
7091:
7086:
7081:
7072:
7071:
7069:
7066:or halves of
7065:
7061:
7057:
7049:
7045:
7042:
7024:
7020:
7016:
7011:
7006:
7001:
6992:
6991:
6990:
6989:
6985:
6984:
6965:
6961:
6957:
6952:
6949:
6940:
6939:
6938:
6937:
6933:
6932:
6931:
6924:
6921:
6908:
6905:
6898:
6894:
6890:
6885:
6878:
6874:
6870:
6865:
6858:
6854:
6850:
6845:
6838:
6834:
6830:
6825:
6820:
6817:
6812:
6809:
6806:
6777:
6773:
6766:
6763:
6756:
6752:
6748:
6743:
6736:
6732:
6728:
6723:
6718:
6715:
6710:
6707:
6700:
6699:
6698:
6688:
6678:
6676:
6670:
6662:
6644:
6641:
6636:
6633:
6630:
6627:
6622:
6619:
6614:
6611:
6608:
6605:
6600:
6597:
6592:
6589:
6586:
6583:
6578:
6575:
6570:
6567:
6564:
6561:
6556:
6553:
6544:
6543:
6540: (1982))
6539:
6521:
6518:
6513:
6510:
6507:
6504:
6499:
6496:
6491:
6488:
6485:
6482:
6477:
6474:
6469:
6466:
6463:
6460:
6455:
6452:
6447:
6444:
6441:
6438:
6433:
6430:
6421:
6420:
6419:
6416:
6415:supercomputer
6413:
6409:
6405:
6401:
6380:
6378:
6370:
6353:
6349:
6344:
6340:
6318:
6314:
6310:
6300:
6296:
6292:
6289:
6283:
6277:
6271:
6243:
6238:
6235:
6232:
6228:
6224:
6219:
6216:
6213:
6209:
6205:
6202:
6194:
6191:
6188:
6184:
6178:
6175:
6172:
6169:
6165:
6161:
6156:
6146:
6143:
6140:
6136:
6132:
6129:
6121:
6117:
6113:
6108:
6105:
6102:
6098:
6091:
6077:
6073:
6066:
6063:
6060:
6053:
6041:
6037:
6030:
6027:
6024:
6018:
6013:
6010:
6007:
6003:
5995:
5994:
5993:
5977:
5972:
5969:
5966:
5963:
5958:
5954:
5947:
5944:
5941:
5936:
5931:
5926:
5922:
5912:
5910:
5909:Peter Borwein
5906:
5902:
5898:
5884:
5878:
5854:
5850:
5846:
5843:
5840:
5837:
5833:
5827:
5819:
5816:
5810:
5804:
5801:
5790:
5787:
5784:
5781:
5775:
5769:
5766:
5758:
5750:
5747:
5731:
5728:
5725:
5721:
5717:
5714:
5709:
5706:
5697:
5696:
5695:
5693:
5689:
5666:
5663:
5659:
5653:
5645:
5642:
5631:
5628:
5625:
5622:
5616:
5610:
5607:
5591:
5588:
5585:
5581:
5575:
5569:
5564:
5558:
5553:
5550:
5541:
5540:
5539:
5537:
5526:
5510:
5507:
5502:
5499:
5496:
5493:
5488:
5485:
5480:
5477:
5474:
5471:
5467:
5459:
5458:
5457:
5455:
5448:
5440:
5432:
5413:
5408:
5403:
5399:
5396:
5393:
5389:
5383:
5380:
5375:
5372:
5368:
5362:
5359:
5354:
5351:
5347:
5341:
5338:
5333:
5330:
5327:
5319:
5313:
5310:
5307:
5304:
5287:
5283:
5279:
5274:
5270:
5251:
5248:
5245:
5241:
5237:
5231:
5228:
5222:
5219:
5216:
5213:
5205:
5202:
5189:
5186:
5183:
5179:
5175:
5173:
5166:
5163:
5154:
5151:
5143:
5133:
5129:
5125:
5122:
5113:
5109:
5100:
5097:
5094:
5089:
5086:
5083:
5077:
5069:
5059:
5055:
5051:
5048:
5039:
5035:
5027:
5024:
5019:
5011:
5007:
5003:
5000:
4996:
4991:
4983:
4973:
4969:
4965:
4962:
4955:
4952:
4946:
4943:
4940:
4937:
4927:
4924:
4920:
4915:
4912:
4906:
4903:
4887:
4884:
4881:
4877:
4868:
4864:
4860:
4857:
4853:
4848:
4846:
4841:
4838:
4835:
4824:
4823:
4822:
4820:
4801:
4797:
4794:
4786:
4782:
4778:
4775:
4771:
4766:
4758:
4754:
4750:
4747:
4743:
4738:
4730:
4726:
4722:
4719:
4715:
4710:
4702:
4698:
4694:
4691:
4687:
4681:
4675:
4670:
4664:
4661:
4658:
4655:
4648:
4638:
4635:
4630:
4614:
4611:
4608:
4604:
4598:
4593:
4587:
4584:
4581:
4578:
4571:
4568:
4560:
4557:
4541:
4538:
4535:
4531:
4525:
4520:
4517:
4510:
4509:
4508:
4506:
4484:
4481:
4479:
4453:
4450:
4447:
4442:
4439:
4434:
4431:
4426:
4423:
4418:
4415:
4410:
4407:
4402:
4399:
4394:
4391:
4386:
4383:
4378:
4375:
4373:
4368:
4357:
4356:
4355:
4353:
4349:
4345:
4333:
4331:
4311:
4308:
4303:
4300:
4297:
4294:
4289:
4286:
4281:
4278:
4275:
4272:
4267:
4264:
4259:
4256:
4253:
4250:
4245:
4242:
4237:
4234:
4231:
4228:
4223:
4220:
4211:
4210:
4209:
4190:
4187:
4182:
4179:
4176:
4173:
4168:
4165:
4160:
4157:
4154:
4151:
4146:
4143:
4138:
4135:
4132:
4129:
4124:
4121:
4116:
4113:
4110:
4107:
4102:
4099:
4090:
4089:
4088:
4085:
4066:
4063:
4058:
4055:
4052:
4049:
4044:
4041:
4036:
4033:
4030:
4027:
4022:
4019:
4014:
4011:
4008:
4005:
4000:
3997:
3988:
3987:
3986:
3967:
3964:
3959:
3956:
3953:
3948:
3945:
3940:
3937:
3934:
3931:
3926:
3923:
3918:
3915:
3912:
3909:
3904:
3901:
3892:
3891:
3890:
3884:
3879:
3873:
3872:
3866:
3857:
3856:Pell equation
3844:
3831:
3825:
3822:
3819:
3811:
3807:
3803:
3798:
3794:
3790:
3784:
3781:
3778:
3772:
3767:
3759:
3756:
3753:
3742:
3741:, producing:
3740:
3736:
3732:
3728:
3724:
3723:Taylor series
3703:
3700:
3695:
3692:
3689:
3684:
3681:
3676:
3673:
3670:
3667:
3662:
3659:
3650:
3649:
3648:
3646:
3640:
3630:
3619:
3617:
3613:
3609:
3600:
3598:
3595:given in the
3590:
3584:
3582:
3578:
3559:
3554:
3551:
3546:
3543:
3540:
3537:
3532:
3529:
3524:
3517:
3516:
3515:
3509:
3500:
3493:
3486:
3479:
3472:
3465:
3458:
3438:
3431:
3428:
3424:
3418:
3414:
3408:
3403:
3400:
3396:
3390:
3382:
3378:
3374:
3369:
3365:
3357:
3353:
3347:
3343:
3339:
3333:
3328:
3325:
3321:
3313:
3312:
3311:
3308:
3301:
3297:
3290:
3286:
3276:
3265:
3255:
3253:
3249:
3245:
3241:
3224:
3221:
3218:
3214:
3210:
3206:
3202:
3198:
3194:
3190:
3186:
3183:
3182:
3181:
3179:
3175:
3170:
3168:
3164:
3160:
3156:
3152:
3144:
3138:
3128:
3126:
3110:
3101:
3096:
3094:
3088:
3078:
3074:
3069:
3063:
3058:
3041:
3039:
3029:
3025:
3005:
2974:
2967:
2964:
2956:
2945:
2941:
2938:
2935:
2927:
2923:
2912:
2905:
2898:
2895:
2888:
2884:
2877:
2873:
2866:
2862:
2861:
2855:
2845:
2841:
2833:
2825:
2821:
2817:
2798:
2790:
2786:
2782:
2779:
2776:
2773:
2769:
2763:
2755:
2752:
2746:
2740:
2737:
2726:
2723:
2720:
2717:
2711:
2705:
2702:
2694:
2686:
2683:
2667:
2664:
2661:
2657:
2653:
2650:
2645:
2642:
2633:
2632:
2631:
2625:
2622:
2621:supercomputer
2614:
2611:In 1989, the
2609:
2595:
2591:
2590:Daniel Shanks
2575:
2573:
2570:, found that
2569:
2564:
2562:
2558:
2549:
2547:
2528:
2525:
2517:
2512:
2508:
2503:
2500:
2493:
2492:
2491:
2461:
2458:
2454:
2448:
2440:
2437:
2426:
2423:
2420:
2417:
2411:
2405:
2402:
2386:
2383:
2380:
2376:
2370:
2364:
2359:
2353:
2348:
2345:
2336:
2335:
2334:
2328:
2322:
2312:
2301:
2296:
2289:
2279:
2278:Planck length
2275:
2265:
2263:
2259:
2255:
2251:
2247:
2227:
2224:
2221:
2218:
2215:
2212:
2209:
2206:
2203:
2200:
2194:
2191:
2181:
2177:
2173:
2172:Taylor series
2169:
2165:
2160:
2147:
2143:
2131:
2128:
2123:
2118:
2115:
2103:
2091:
2088:
2083:
2078:
2075:
2063:
2051:
2048:
2043:
2038:
2035:
2023:
2011:
2008:
2003:
1998:
1995:
1983:
1979:
1972:
1969:
1966:
1963:
1958:
1955:
1949:
1943:
1940:
1937:
1934:
1929:
1926:
1919:
1908:
1905:
1902:
1898:
1894:
1888:
1885:
1880:
1876:
1872:
1865:
1861:
1857:
1844:
1841:
1838:
1834:
1830:
1825:
1822:
1812:
1810:
1806:
1801:
1799:
1791:
1787:
1783:
1782:Cyclometricus
1778:
1776:
1768:
1764:
1759:
1757:
1749:
1739:
1737:
1718:
1715:
1712:
1705:
1704:
1703:
1686:
1683:
1680:
1677:
1674:
1667:
1666:
1665:
1663:
1655:
1654:mathematician
1651:
1648:(KΔshΔnΔ«), a
1647:
1643:
1615:
1612:
1609:
1604:
1600:
1596:
1591:
1588:
1583:
1579:
1572:
1566:
1563:
1560:
1557:
1550:
1542:
1539:
1530:
1527:
1524:
1519:
1516:
1511:
1506:
1503:
1498:
1493:
1490:
1485:
1482:
1479:
1476:
1472:
1468:
1461:
1460:
1459:
1456:
1434:
1430:
1426:
1407:
1388:
1384:
1381:
1373:
1369:
1365:
1362:
1358:
1353:
1345:
1341:
1337:
1334:
1330:
1325:
1319:
1316:
1313:
1309:
1304:
1301:
1297:
1291:
1286:
1280:
1277:
1274:
1271:
1264:
1254:
1251:
1246:
1230:
1227:
1224:
1220:
1214:
1209:
1203:
1200:
1197:
1194:
1187:
1184:
1176:
1173:
1157:
1154:
1151:
1147:
1141:
1136:
1133:
1126:
1125:
1124:
1111:
1103:
1097:
1093:
1087:
1084:
1081:
1078:
1075:
1051:
1047:
1044:
1039:
1036:
1031:
1026:
1023:
1018:
1013:
1010:
1005:
1002:
998:
994:
991:
988:
981:
980:
979:
966:
960:
954:
951:
948:
945:
942:
934:
926:
922:
918:
914:
904:
902:
898:
894:
890:
888:
864:
863:
856:
852:
850:
846:
842:
837:
810:
799:
760:
757:
752:
750:
734:
729:
725:
720:
716:
713:
710:
706:
702:
699:
696:
678:
673:
671:
643:
635:
633:
629:
628:Bhishma Parva
621:
618:
610:
605:
603:
602:Bhishma Parva
599:
594:
579:
578:
572:
570:
552:
548:
544:
539:
523:
519:
515:
507:
503:
501:
483:
478:
476:
472:
461:Early history
458:
452:
448:
437:
428:
426:
422:
418:
414:
409:
407:
403:
399:
391:
388:
384:
373:
368:
366:
361:
359:
354:
353:
351:
350:
345:
339:
337:
333:Six nines in
331:
329:
328:Basel problem
326:
324:
321:
320:
319:
318:
314:
313:
308:
305:
303:
300:
299:
298:
297:
293:
292:
287:
286:
282:
280:
277:
276:
275:
274:
270:
269:
264:
261:
259:
256:
254:
251:
249:
246:
244:
241:
239:
236:
234:
233:William Jones
231:
229:
226:
224:
223:Seki Takakazu
221:
219:
216:
214:
211:
209:
206:
204:
201:
199:
196:
194:
191:
189:
186:
184:
181:
180:
179:
178:
174:
173:
168:
165:
163:
160:
158:
155:
153:
150:
149:
148:
147:
143:
142:
137:
136:Transcendence
134:
132:
131:Irrationality
129:
128:
127:
126:
122:
121:
116:
113:
111:
110:Circumference
108:
106:
103:
102:
101:
100:
96:
95:
92:
75:
74:
71:
65:
64:
60:
56:
55:
48:
44:
36:
29:
22332:
22313:
22294:
22275:
22244:
22213:
22209:
22169:. Retrieved
22165:the original
22154:
22141:
22129:. Retrieved
22119:
22107:. Retrieved
22097:
22085:. Retrieved
22081:the original
22076:
22067:
22055:
22034:
22022:
22009:
21985:
21981:
21975:
21942:
21938:
21928:
21914:
21902:
21889:
21877:
21864:
21845:
21839:
21831:
21827:
21822:
21803:
21769:
21763:
21754:
21750:
21740:
21713:
21709:
21703:
21678:
21674:
21640:
21635:
21625:
21604:
21585:
21576:
21567:
21557:
21547:
21540:
21530:
21523:
21478:
21442:
21438:
21435:
21425:
21421:
21411:
21399:
21389:
21383:
21373:
21367:
21360:
21350:
21344:
21341:
21336:
21324:
21297:
21292:, p. 66
21285:
21273:
21262:
21250:. Retrieved
21235:
21223:. Retrieved
21219:the original
21208:
21198:
21190:
21178:. Retrieved
21159:(1): 75β84.
21156:
21152:
21135:
21118:
21114:
21108:
21096:. Retrieved
21089:the original
21084:
21072:
21060:. Retrieved
21056:
21046:
21034:. Retrieved
21030:
21020:
21008:. Retrieved
21004:
20994:
20982:. Retrieved
20978:
20968:
20956:. Retrieved
20949:
20940:
20928:. Retrieved
20923:The Guardian
20921:
20912:
20900:. Retrieved
20891:
20879:. Retrieved
20869:
20857:. Retrieved
20848:
20836:. Retrieved
20832:
20823:
20802:
20790:. Retrieved
20786:
20736:
20724:. Retrieved
20720:
20710:
20696:
20678:
20664:
20652:. Retrieved
20648:the original
20643:
20634:
20609:
20605:
20563:
20551:. Retrieved
20544:
20535:
20506:
20502:
20492:
20474:
20467:
20424:
20418:
20415:"Value of Ο"
20394:
20385:
20376:
20370:the original
20339:
20335:
20325:
20320:
20316:
20298:
20265:
20261:
20251:
20241:
20229:
20218:
20212:
20208:
20203:
20199:
19990:
19986:
19982:
19978:
19974:
19972:
19959:
19949:
19940:
19936:
19923:
19911:. Retrieved
19898:
19882:(2): 64β85.
19879:
19875:
19865:
19852:
19842:
19833:
19827:
19818:
19814:
19808:
19795:
19759:
19753:
19734:
19728:
19716:. Retrieved
19702:
19689:
19652:
19634:
19613:
19587:
19583:
19577:
19569:the original
19564:
19554:
19533:
19498:
19494:
19484:
19471:
19448:
19439:
19426:
19417:
19411:
19393:
19381:
19362:
19352:
19323:
19311:. Retrieved
19307:the original
19301:
19294:
19271:
19246:
19228:
19219:
19210:
19204:
19192:. Retrieved
19188:
19178:
19159:
19092:
19061:
19055:
19046:overclocking
19028:
19014:
19007:
18994:
18991:
18985:
18968:
18936:
18921:
18901:
18882:
18641:
18455:
18448:
18025:
17700:
17679:
17663:
17661:
17464:
17445:
17430:
17091:
17087:
17084:
16971:
16967:
16963:
16944:
16748:
16740:
16158:
16098:
16064:
16035:
15936:
15745:
15682:
15323:
15144:
14860:
14730:
14605:
13838:
13777:
13770:
13587:
13579:Trigonometry
13550:
13409:
13091:
12512:
12500:
12493:
12364:
12360:
12358:
11930:
11917:
11911:
11905:
11899:
11892:
11673:
11662:
11600:
11594:
11576:
11570:
11564:
11558:
11555:
11493:
11487:
11476:
11470:
11468:
11423:
11069:
11065:
11061:
11021:modular form
10627:
10531:
10337:
10225:
10221:
10219:class number
10214:
10080:
9754:
9475:
9468:
9461:
9454:
6925:
6922:
6794:
6684:
6666:
6381:
6377:access times
6263:
5913:
5890:
5874:
5686:
5534:
5452:
4817:
4503:
4346:
4339:
4328:
4207:
4086:
4084:as a check.
4083:
3984:
3880:
3869:
3867:
3865:= β1.)
3845:
3743:
3730:
3720:
3642:
3620:
3601:
3596:
3585:
3580:
3574:
3507:
3498:
3491:
3484:
3477:
3470:
3463:
3456:
3453:
3306:
3299:
3295:
3288:
3284:
3274:
3272:
3250:flower or a
3236:
3209:handbreadths
3204:
3200:
3188:
3171:
3151:1 Kings 7:23
3143:Hebrew Bible
3140:
3108:
3097:
3090:
3081:Indiana bill
3072:
3068:Hebrew Bible
3061:
3050:
3035:
3021:
2987:
2934:Google Cloud
2813:
2610:
2576:
2565:
2550:
2543:
2481:
2333:, including
2324:
2297:
2266:
2180:the identity
2161:
1813:
1802:
1781:
1779:
1766:
1760:
1745:
1733:
1701:
1644:
1633:
1457:
1437:
1428:
1421:
1067:
919:, found the
910:
884:
868:
860:
854:
838:
753:
674:
639:
626:
619:
611:
607:
595:
575:
573:
540:
518:Hebrew Bible
504:
479:
464:
429:
410:
381:
283:
228:Takebe Kenko
167:Memorization
156:
76:
43:
22087:11 December
22077:crd.LBL.gov
22061:Bellard.org
22046:0912.0303v1
21568:Extremetech
20654:11 December
20549:. July 2007
20513:: 318β319.
20509:(139β147).
20268:(1): 1β14.
20214:John Machin
19983:Curve Lines
19233:Grove Press
18908:many digits
17706:his formula
17672:hexadecimal
16953:(using base
15743:increases.
11064:= 3502 has
10334:j-invariant
9165:11222.11122
7070:triangles.
7068:equilateral
7044:Karl Popper
6697:, which is
3159:round basin
2818:. In 1999,
2557:calculators
2302:calculated
2274:light-years
2258:John Machin
2164:John Machin
1805:John Wallis
1662:sexagesimal
935:, based on
907:Middle Ages
862:ΔryabhaαΉΔ«ya
849:ΔryabhaαΉΔ«ya
798:Zu Chongzhi
765:to between
749:sexagesimal
632:Mahabharata
598:Mahabharata
500:Old Kingdom
434:is held by
253:John Wrench
238:John Machin
193:Zu Chongzhi
32:history of
23:; see also
22354:Categories
22184:References
22126:"TachusPi"
21834:(2009) 399
21643:(4): 554.
21637:Phys. Rev.
21488:1802.07558
21252:30 October
21225:30 October
20881:30 January
20859:30 January
20814:1612.00489
20594:Shanks, D.
20553:22 January
20358:2008467244
20220:Van Ceulen
19913:13 January
19903:Capra, B.
19686:= 3.1416,
19545:2008.07995
14917:such that
13829:Arctangent
12361:exactly on
11368:Of these,
9791:1777729635
9780:3949122332
9303:8769956796
4350:(see also
3223:Maimonides
3135:See also:
2963:y-cruncher
2887:y-cruncher
2529:3.14159273
2254:Jurij Vega
2176:arctangent
642:Archimedes
469:dating to
457:) digits.
451:y-cruncher
425:Jurij Vega
402:Common Era
294:In culture
279:Chronology
183:Archimedes
123:Properties
22024:MathWorld
22002:123395287
21959:0002-9890
21695:121628039
21515:214742997
21459:123395287
21416:(1798) .
21317:Eves 1992
21302:Eves 1992
21278:Eves 1992
21175:0315-0860
20930:31 August
20902:31 August
20527:120851313
20451:1476-4687
20366:448882242
20290:121087222
20175:&
20168:−
20163:¯
20140:−
20098:¯
20075:−
20038:−
20033:¯
20017:−
19525:146074061
19517:2550-0651
19501:(1): 18.
19430:See also
19227:(2001) .
19023:Pentium 4
18998:-cruncher
18943:libraries
18827:
18750:
18695:
18648:Algorithm
18607:−
18559:545140134
18527:∞
18512:∑
18484:π
18456:In 1988,
18353:∞
18338:∑
18310:π
18261:⋯
18132:∞
18117:∑
18070:∞
18055:∑
18043:π
18030:include:
17954:−
17923:−
17892:−
17837:−
17806:−
17770:−
17758:∞
17743:∑
17719:π
17674:digit of
17618:−
17594:−
17570:−
17522:∞
17507:∑
17500:π
17465:In 1997,
17354:−
17323:−
17292:−
17237:−
17206:−
17169:−
17158:∞
17143:∑
17119:π
17019:∞
17004:∑
16991:π
16879:−
16855:−
16831:−
16800:∞
16785:∑
16778:π
16722:⋯
16714:167772160
16552:⋅
16541:∞
16526:∑
16518:⋯
16506:⋅
16478:⋅
16450:⋅
16422:⋅
16387:⋯
16368:⋅
16362:⋅
16356:⋅
16350:⋅
16339:⋅
16333:⋅
16308:⋅
16302:⋅
16296:⋅
16285:⋅
16260:⋅
16254:⋅
16200:
16192:−
16174:π
16122:π
16113:
16079:π
16000:−
15989:−
15980:
15969:≥
15962:∑
15950:π
15922:⋯
15906:
15887:
15868:
15849:
15814:
15806:∞
15791:∑
15779:π
15754:π
15711:π
15691:π
15639:−
15625:−
15599:−
15573:−
15510:−
15470:−
15444:−
15211:−
15193:∞
15178:∑
15159:
15003:∞
14988:∑
14972:
14898:−
14840:≥
14811:−
14800:−
14791:
14764:π
14739:π
14692:⋯
14617:π
14587:⋯
14366:∞
14351:∑
14270:∞
14255:∑
14194:∞
14179:∑
14157:⋯
14143:⋅
14137:⋅
14131:⋅
14125:⋅
14107:⋅
14101:⋅
14095:⋅
14089:⋅
14064:⋅
14058:⋅
14052:⋅
14034:⋅
14028:⋅
14022:⋅
13997:⋅
13991:⋅
13973:⋅
13967:⋅
13942:⋅
13924:⋅
13858:π
13809:π
13751:⋯
13748:−
13732:−
13706:−
13643:−
13627:∞
13612:∑
13602:π
13392:…
13341:⋱
13199:−
13173:−
13157:−
13143:−
13117:∑
13107:π
13052:⋱
12850:⋱
12686:π
12648:⋱
12536:π
12521:has many
12488:3.141549
11818:≤
11760:−
11750:∑
11735:−
11725:∑
11702:∞
11699:→
11685:π
11642:≤
11584:between β
11568:), where
11441:π
11043:−
11035:τ
10777:−
10745:−
10706:−
10667:−
10563:
10481:−
10376:−
10365:
10257:
10113:
9992:−
9956:
9700:−
9681:−
9659:−
9573:165707065
9406:−
9385:−
9237:888582403
9061:×
8962:Ramanujan
8773:−
8737:−
8709:−
8435:−
8281:⋅
8275:⋅
8264:⋅
8258:⋅
8137:−
7953:−
7922:−
7896:−
7832:−
7733:−
7717:−
7516:Ramanujan
7346:−
7292:−
7217:γ
7157:γ
7154:−
7087:−
7064:isosceles
6846:−
6807:≊
6669:normality
6637:
6615:
6606:−
6593:
6571:
6554:π
6514:
6492:
6483:−
6470:
6448:
6431:π
6293:−
6162:−
6028:−
5970:−
5942:−
5788:545140134
5748:−
5737:∞
5722:∑
5710:π
5597:∞
5582:∑
5554:π
5536:Ramanujan
5503:
5481:
5468:π
5400:⋯
5257:∞
5242:∑
5195:∞
5180:∑
5164:π
5155:⋯
5098:⋅
5087:⋅
4893:∞
4878:∑
4839:
4798:⋯
4779:⋅
4767:−
4751:⋅
4723:⋅
4711:−
4695:⋅
4631:−
4620:∞
4605:∑
4569:−
4558:−
4547:∞
4532:∑
4518:π
4482:≈
4387:−
4376:≈
4369:π
4344:include:
4304:
4282:
4273:−
4260:
4238:
4221:π
4183:
4161:
4152:−
4139:
4117:
4100:π
4059:
4050:−
4037:
4015:
3998:π
3960:
3941:
3919:
3902:π
3804:⋅
3782:−
3773:⋅
3696:
3690:−
3677:
3660:π
3541:π
2826:computed
2724:545140134
2684:−
2673:∞
2658:∑
2646:π
2615:computed
2561:computers
2526:≈
2504:≈
2501:π
2392:∞
2377:∑
2349:π
2256:improved
2225:
2219:−
2213:
2201:π
2162:In 1706,
2148:⋯
2144:⋅
2124:⋅
2104:⋅
2084:⋅
2064:⋅
2044:⋅
2024:⋅
2004:⋅
1950:⋅
1941:−
1914:∞
1899:∏
1886:−
1850:∞
1835:∏
1823:π
1803:In 1656,
1775:tombstone
1754:known as
1716:≈
1713:π
1681:≈
1678:π
1573:±
1564:−
1540:−
1531:−
1528:⋯
1512:−
1486:−
1480:≈
1469:π
1385:⋯
1366:⋅
1354:−
1338:⋅
1317:⋅
1305:−
1247:−
1236:∞
1221:∑
1185:−
1174:−
1163:∞
1148:∑
1134:π
1088:
1076:π
1048:⋯
1032:−
1006:−
989:π
955:
943:π
845:Aryabhata
396:) in the
198:Aryabhata
22242:(1971).
22131:20 March
22109:18 April
21615:Archived
21584:(1995).
21394:(1755).
21378:(1971).
21246:Archived
21098:16 March
21062:16 March
21036:16 March
20984:16 March
20838:14 March
20792:14 March
20726:18 April
20686:Archived
20323:: 41β44.
20282:41133896
19957:(1706).
19853:MacTutor
19796:MacTutor
19642:Archived
19360:(1993).
19107:See also
19094:Super PI
18986:TachusPi
18874:Projects
18553:13591409
16701:54525952
14958:formula
12422:3.22222
11834:if
11787:if
11582:integers
9576:52746197
8993:28658146
5905:Jonathan
5782:13591409
4332:(1896).
3645:Machin's
3597:Almagest
3197:geometry
3195:text on
2718:13591409
2624:IBM 3090
2286:10
2272:billion
1784:(1621),
858:β
851:stated:
821:and Ο β
751:digits.
623:β
385:for the
89:26433...
57:Part of
22267:0449960
22218:Bibcode
21967:2975006
21897:(ed.).
21872:(ed.).
21645:Bibcode
21396:"Β§2.30"
21382:(ed.).
21180:14 July
20958:10 June
20626:2003813
20459:4085398
20429:Bibcode
20207:. This
19975:Lengths
19718:20 July
19681:⁄
19606:0875525
19565:Zimaths
19128:Pi is 3
19078:√
19068:√
19056:QuickPi
19037:√
17694:bit of
16760:base 16
16688:2883584
16159:yields
16095:Arcsine
16063:is the
13566:⁄
13556:⁄
12485:3141549
12477:3.1417
12466:3.1425
12433:3.0625
12376:⁄
12339:(5,β5)
12321:(β1,β5)
12318:(β2,β5)
12315:(β3,β5)
12312:(β4,β5)
12309:(β5,β5)
12304:(5,β4)
12286:(β1,β4)
12283:(β2,β4)
12280:(β3,β4)
12277:(β4,β4)
12274:(β5,β4)
12269:(5,β3)
12251:(β1,β3)
12248:(β2,β3)
12245:(β3,β3)
12242:(β4,β3)
12239:(β5,β3)
12234:(5,β2)
12216:(β1,β2)
12213:(β2,β2)
12210:(β3,β2)
12207:(β4,β2)
12204:(β5,β2)
12199:(5,β1)
12181:(β1,β1)
12178:(β2,β1)
12175:(β3,β1)
12172:(β4,β1)
12169:(β5,β1)
11903:,
11598:,
11562:,
11491:,
11350:1725033
11347:5419351
11337:1360120
11334:4272943
11321:1146408
11072:) = 16.
10149:3.14159
10025:3.14159
9879:3.14159
9799:3.14159
9711:3.14159
9584:3.14159
9524:3.14159
9417:3.14159
9316:3.14159
9250:3.14159
9179:3.14159
9107:3.14159
9006:3.14159
8931:3.14159
8785:3.14159
8657:3.14159
8610:3.14159
8527:3.14159
8454:3.14159
8380:3.14159
8212:3.14159
8156:3.14159
8093:3.14159
8035:3.14159
7851:3.14159
7766:3.14159
7670:3.14155
7627:3.14156
7414:3.14142
7359:3.14142
7305:3.14142
7205:is the
6767:3.14159
6412:Hitachi
6391:⁄
5529:arctan.
5437:is the
4505:Madhava
4348:Liu Hui
3581:Metrica
3201:outside
3176:and in
3116:⁄
3011:⁄
2997:⁄
2854:days).
1796:from a
1411:√
878:⁄
826:⁄
816:⁄
791:⁄
759:Liu Hui
684:⁄
677:Ptolemy
670:96-gons
663:⁄
649:⁄
630:of the
591:β 3.139
586:⁄
569:octagon
562:⁄
533:⁄
516:in the
493:⁄
271:History
203:Madhava
188:Liu Hui
78:3.14159
22339:
22320:
22301:
22282:
22265:
22255:
22197:&
22000:
21965:
21957:
21852:
21810:
21776:
21693:
21592:
21513:
21503:
21457:
21173:
21010:2 July
20624:
20525:
20457:
20449:
20420:Nature
20364:
20356:
20346:
20288:
20280:
20209:Series
19991:Circle
19987:Planes
19770:
19741:
19621:
19604:
19523:
19515:
19459:
19374:
19313:7 June
19282:
19253:cubits
19239:
19194:2 July
19074:, and
19015:PiFast
18884:Pi Hex
18879:Pi Hex
18610:640320
18498:426880
18291:Newton
17686:. The
17668:binary
17485:as an
17458:; the
17102:
16959:
16955:
16036:where
15977:arctan
15903:arctan
15884:arctan
15865:arctan
15846:arctan
15811:arctan
15324:where
15156:arctan
14969:arctan
14861:where
14788:arctan
14579:230945
14566:109395
13773:arctan
12336:(4,β5)
12333:(3,β5)
12330:(2,β5)
12327:(1,β5)
12324:(0,β5)
12301:(4,β4)
12298:(3,β4)
12295:(2,β4)
12292:(1,β4)
12289:(0,β4)
12266:(4,β3)
12263:(3,β3)
12260:(2,β3)
12257:(1,β3)
12254:(0,β3)
12231:(4,β2)
12228:(3,β2)
12225:(2,β2)
12222:(1,β2)
12219:(0,β2)
12196:(4,β1)
12193:(3,β1)
12190:(2,β1)
12187:(1,β1)
12184:(0,β1)
12164:(5,0)
12146:(β1,0)
12143:(β2,0)
12140:(β3,0)
12137:(β4,0)
12134:(β5,0)
12129:(5,1)
12111:(β1,1)
12108:(β2,1)
12105:(β3,1)
12102:(β4,1)
12099:(β5,1)
12094:(5,2)
12076:(β1,2)
12073:(β2,2)
12070:(β3,2)
12067:(β4,2)
12064:(β5,2)
12059:(5,3)
12041:(β1,3)
12038:(β2,3)
12035:(β3,3)
12032:(β4,3)
12029:(β5,3)
12024:(5,4)
12006:(β1,4)
12003:(β2,4)
12000:(β3,4)
11997:(β4,4)
11994:(β5,4)
11989:(5,5)
11971:(β1,5)
11968:(β2,5)
11965:(β3,5)
11962:(β4,5)
11959:(β5,5)
11324:364913
11311:265381
11308:833719
11295:312689
11282:208341
11269:104348
11256:103993
10626:where
10276:236674
10176:
10170:
10164:
10158:
10152:
10120:640320
10052:
10046:
10040:
10034:
10028:
9900:
9894:
9888:
9882:
9814:
9808:
9802:
9758:= 253.
9726:
9720:
9714:
9599:
9593:
9587:
9539:
9533:
9527:
9426:
9420:
9325:
9319:
9259:
9253:
9182:
9110:
9009:
8934:
8788:
8660:
8613:
8530:
8457:
8383:
8215:
8159:
8096:
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16873:5
16870:+
16867:n
16864:8
16860:1
16849:4
16846:+
16843:n
16840:8
16836:2
16825:1
16822:+
16819:n
16816:8
16812:4
16806:(
16795:0
16792:=
16789:n
16781:=
16765:Ο
16755:Ο
16719:+
16706:+
16693:+
16680:+
16667:+
16654:+
16646:9
16641:+
16636:8
16633:1
16628:+
16625:3
16622:=
16609:)
16606:1
16603:+
16600:n
16597:2
16594:(
16589:n
16576:)
16571:n
16567:n
16564:2
16558:(
16549:3
16536:0
16533:=
16530:n
16522:=
16515:+
16509:7
16501:3
16487:+
16481:5
16473:2
16459:+
16453:3
16445:1
16436:6
16431:+
16425:1
16417:0
16408:3
16403:=
16392:)
16384:+
16376:7
16372:2
16365:7
16359:6
16353:4
16347:2
16342:5
16336:3
16330:1
16324:+
16316:5
16312:2
16305:5
16299:4
16293:2
16288:3
16282:1
16276:+
16268:3
16264:2
16257:3
16251:2
16247:1
16242:+
16237:2
16234:1
16228:(
16224:6
16221:=
16217:)
16212:2
16209:1
16204:(
16195:1
16184:6
16181:=
16142:2
16139:1
16134:=
16130:)
16125:6
16117:(
16065:n
16049:n
16045:F
16021:,
16014:k
16010:a
16003:1
15997:k
15993:a
15986:2
15972:2
15966:k
15958:=
15953:4
15919:+
15911:1
15900:+
15895:5
15892:1
15881:+
15876:2
15873:1
15862:+
15857:1
15854:1
15843:=
15836:1
15833:+
15830:n
15827:2
15823:F
15819:1
15801:0
15798:=
15795:n
15787:=
15782:2
15731:k
15664:,
15661:x
15657:/
15653:)
15650:x
15647:(
15642:1
15636:n
15632:a
15628:4
15621:)
15615:2
15611:x
15606:/
15602:4
15596:1
15592:(
15587:)
15584:x
15581:(
15576:1
15570:n
15566:b
15562:=
15559:)
15556:x
15553:(
15548:n
15544:b
15535:,
15532:x
15528:/
15524:)
15521:x
15518:(
15513:1
15507:n
15503:b
15499:4
15496:+
15492:)
15486:2
15482:x
15477:/
15473:4
15467:1
15463:(
15458:)
15455:x
15452:(
15447:1
15441:n
15437:a
15433:=
15430:)
15427:x
15424:(
15419:n
15415:a
15406:,
15403:1
15400:=
15397:)
15394:x
15391:(
15386:1
15382:b
15373:,
15370:x
15366:/
15362:2
15359:=
15356:)
15353:x
15350:(
15345:1
15341:a
15309:,
15301:)
15298:x
15295:(
15289:2
15284:n
15280:b
15276:+
15272:)
15269:x
15266:(
15260:2
15255:n
15251:a
15244:)
15241:x
15238:(
15231:n
15226:a
15214:1
15208:n
15205:2
15201:1
15188:1
15185:=
15182:n
15174:2
15171:=
15168:)
15165:x
15162:(
15130:.
15122:1
15119:+
15116:n
15112:)
15106:2
15102:x
15098:+
15095:1
15092:(
15086:1
15083:+
15080:n
15077:2
15073:x
15063:!
15060:)
15057:1
15054:+
15051:n
15048:2
15045:(
15038:2
15034:)
15030:!
15027:n
15024:(
15019:n
15016:2
15012:2
14998:0
14995:=
14992:n
14984:=
14981:)
14978:x
14975:(
14940:2
14935:=
14930:1
14926:a
14901:1
14895:k
14891:a
14887:+
14884:2
14879:=
14874:k
14870:a
14846:,
14843:2
14837:k
14832:,
14825:k
14821:a
14814:1
14808:k
14804:a
14797:2
14785:=
14778:1
14775:+
14772:k
14768:2
14706:)
14701:)
14696:)
14689:+
14686:2
14682:(
14676:7
14673:3
14668:+
14665:2
14661:(
14655:5
14652:2
14647:+
14644:2
14640:(
14634:3
14631:1
14626:+
14623:2
14620:=
14584:+
14571:+
14558:+
14545:+
14532:+
14519:+
14506:+
14498:4
14493:+
14485:4
14480:+
14475:3
14472:2
14467:+
14464:2
14461:=
14446:)
14443:1
14440:+
14437:n
14434:2
14431:(
14425:)
14420:n
14416:n
14413:2
14407:(
14390:1
14387:+
14384:n
14380:2
14361:0
14358:=
14355:n
14347:=
14339:!
14336:)
14333:1
14330:+
14327:n
14324:2
14321:(
14307:2
14303:!
14299:n
14294:1
14291:+
14288:n
14284:2
14265:0
14262:=
14259:n
14251:=
14243:!
14240:!
14237:)
14234:1
14231:+
14228:n
14225:2
14222:(
14210:!
14207:n
14189:0
14186:=
14183:n
14175:2
14172:=
14161:)
14154:+
14140:9
14134:7
14128:5
14122:3
14110:5
14104:4
14098:3
14092:2
14086:1
14075:+
14067:9
14061:7
14055:5
14049:3
14037:4
14031:3
14025:2
14019:1
14008:+
14000:7
13994:5
13988:3
13976:3
13970:2
13964:1
13953:+
13945:5
13939:3
13927:2
13921:1
13910:+
13902:3
13890:1
13879:+
13876:1
13872:(
13868:2
13865:=
13841:Ο
13789:x
13778:x
13755:)
13745:+
13740:7
13737:1
13727:5
13724:1
13719:+
13714:3
13711:1
13701:1
13698:1
13692:(
13688:4
13685:=
13677:1
13674:+
13671:n
13668:2
13654:n
13650:)
13646:1
13640:(
13622:0
13619:=
13616:n
13608:4
13605:=
13558:7
13547:.
13532:m
13529:5
13526:+
13521:3
13517:m
13513:4
13508:1
13505:+
13500:2
13496:m
13489:4
13486:=
13474:m
13471:2
13465:2
13461:2
13455:+
13452:m
13449:2
13443:2
13439:1
13433:+
13430:m
13427:2
13423:2
13395:)
13389:,
13386:3
13383:,
13380:2
13377:,
13374:1
13371:=
13368:m
13365:(
13338:+
13335:m
13332:2
13318:2
13314:3
13302:+
13299:m
13296:2
13282:2
13278:2
13266:+
13263:m
13260:2
13246:2
13242:1
13230:+
13227:m
13224:2
13210:m
13206:)
13202:1
13196:(
13193:2
13182:+
13176:1
13170:n
13167:2
13160:1
13154:n
13150:)
13146:1
13140:(
13132:m
13127:1
13124:=
13121:n
13113:4
13110:=
13049:+
13032:2
13028:5
13016:+
12999:2
12995:6
12983:+
12980:9
12966:2
12962:3
12950:+
12947:7
12933:2
12929:4
12917:+
12914:5
12900:2
12896:1
12884:+
12881:3
12878:=
12847:+
12844:9
12830:2
12826:4
12814:+
12811:7
12797:2
12793:3
12781:+
12778:5
12764:2
12760:2
12748:+
12745:3
12731:2
12727:1
12715:+
12712:1
12700:4
12689:=
12645:+
12642:6
12628:2
12624:5
12612:+
12609:6
12595:2
12591:3
12579:+
12576:6
12562:2
12558:1
12546:+
12543:3
12539:=
12519:Ο
12503:Ο
12438:5
12427:4
12416:3
12405:2
12399:Ο
12391:r
12383:r
12378:5
12369:Ο
11949:r
11945:Ο
11941:r
11937:r
11932:r
11927:r
11923:r
11918:y
11912:x
11906:y
11900:x
11895:r
11871:.
11868:r
11858:2
11854:y
11850:+
11845:2
11841:x
11828:0
11821:r
11811:2
11807:y
11803:+
11798:2
11794:x
11781:1
11775:{
11768:r
11763:r
11757:=
11754:y
11743:r
11738:r
11732:=
11729:x
11717:2
11713:r
11709:1
11696:r
11688:=
11669:r
11665:Ο
11648:.
11645:r
11635:2
11631:y
11627:+
11622:2
11618:x
11601:y
11595:x
11590:r
11586:r
11577:y
11571:x
11565:y
11559:x
11541:.
11534:2
11530:y
11526:+
11521:2
11517:x
11511:=
11508:d
11494:y
11488:x
11478:r
11472:r
11454:.
11449:2
11445:r
11438:=
11435:A
11414:Ο
11397:Ο
11342:,
11329:,
11316:,
11303:,
11290:,
11277:,
11264:,
11251:,
11238:,
11225:,
11220:7
11212:,
11207:1
11204:3
11186:Ο
11178:Ο
11134:.
11131:.
11128:.
11107:.
11104:.
11101:.
11070:d
11066:h
11062:d
11038:=
10995:)
10990:2
10982:+
10976:(
10970:2
10967:1
10961:=
10954:d
10947:)
10942:2
10934:+
10928:(
10925:=
10918:c
10911:)
10901:7
10898:+
10893:2
10885:(
10879:2
10876:1
10870:=
10863:b
10856:)
10846:4
10843:+
10837:(
10831:2
10828:1
10822:=
10815:a
10785:)
10780:1
10772:2
10768:d
10762:+
10759:d
10756:(
10753:)
10748:1
10740:2
10736:c
10730:+
10727:c
10724:(
10719:2
10715:)
10709:1
10701:2
10697:b
10691:+
10688:b
10685:(
10680:2
10676:)
10670:1
10662:2
10658:a
10652:+
10649:a
10646:(
10643:=
10640:u
10628:u
10600:)
10592:+
10587:6
10583:)
10579:u
10576:2
10573:(
10568:(
10538:.
10532:G
10501:7
10494:5
10487:)
10472:)
10468:)
10465:3
10462:+
10452:(
10449:)
10439:+
10434:7
10429:(
10426:)
10421:7
10416:+
10411:5
10406:(
10403:)
10398:5
10393:+
10390:3
10387:(
10384:(
10372:2
10368:(
10338:d
10308:)
10302:+
10297:3
10293:)
10279:+
10273:(
10268:3
10260:(
10232:.
10226:d
10222:h
10215:d
10184:+
10146:=
10135:)
10129:+
10124:3
10116:(
10088:.
10081:g
10060:+
10056:9
10022:=
10018:)
10011:+
9999:)
9995:1
9986:4
9982:5
9977:(
9967:2
9960:(
9945:1
9908:+
9876:=
9865:2
9822:+
9796:=
9785:2
9755:n
9734:+
9708:=
9703:1
9695:)
9654:4
9643:(
9607:+
9603:4
9581:=
9547:+
9543:9
9521:=
9510:2
9481:)
9476:Ο
9474:(
9472:4
9469:E
9462:Ο
9460:(
9458:6
9455:E
9434:+
9414:=
9409:1
9401:)
9380:6
9369:(
9333:+
9313:=
9267:+
9263:8
9247:=
9214:Ο
9210:Ο
9190:+
9176:=
9118:+
9104:=
9096:5
9088:+
9085:7
9078:5
9070:+
9017:+
9003:=
8972:.
8970:Ο
8942:+
8928:=
8922:4
8908:=
8902:4
8892:2
8888:)
8882:3
8879:2
8874:(
8871:+
8868:2
8864:1
8859:+
8854:4
8850:2
8846:+
8841:4
8837:3
8796:+
8782:=
8756:2
8745:=
8740:1
8732:)
8720:2
8704:4
8693:(
8668:+
8654:=
8621:+
8607:=
8591:2
8579:2
8572:2
8564:+
8561:3
8538:+
8524:=
8511:2
8499:2
8494:5
8465:+
8451:=
8446:9
8391:+
8377:=
8373:)
8356:2
8352:5
8346:+
8337:2
8333:3
8327:+
8318:2
8314:1
8308:+
8301:(
8295:2
8290:)
8284:7
8278:5
8272:3
8267:6
8261:4
8255:2
8249:(
8223:+
8209:=
8201:2
8167:+
8153:=
8148:8
8104:+
8090:=
8046:+
8042:3
8032:=
8013:2
8009:3
8003:+
7994:2
7990:2
7984:+
7975:2
7971:1
7965:+
7958:2
7949:)
7943:9
7940:1
7935:+
7930:7
7927:1
7917:5
7914:1
7909:+
7904:3
7901:1
7893:1
7889:(
7885:4
7862:+
7858:0
7848:=
7837:3
7827:2
7819:+
7814:3
7809:+
7804:2
7777:+
7773:6
7763:=
7758:2
7753:)
7745:2
7741:2
7736:2
7728:2
7723:2
7714:2
7710:(
7674:+
7666:=
7660:5
7631:+
7623:=
7616:9
7612:4
7606:7
7602:7
7565:+
7557:=
7551:5
7548:9
7542:+
7537:5
7534:9
7494:+
7486:=
7477:5
7472:+
7469:6
7464:+
7461:7
7425:+
7421:1
7411:=
7402:2
7396:+
7393:3
7370:+
7366:8
7356:=
7316:+
7312:2
7302:=
7262:+
7254:=
7248:3
7193:e
7173:,
7168:+
7160:=
7151:e
7148:+
7145:1
7111:+
7103:=
7100:1
7097:+
7092:3
7052:Ο
7025:+
7017:=
7012:3
7007:+
7002:2
6966:+
6958:=
6953:7
6928:Ο
6906:=
6899:8
6886:+
6879:6
6866:+
6859:3
6839:2
6826:+
6818:8
6813:+
6810:3
6799::
6797:Ο
6778:+
6764:=
6757:3
6744:+
6737:2
6724:+
6716:8
6711:+
6708:3
6691:Ο
6674:Ο
6659:(
6642:1
6628:+
6620:1
6598:1
6587:7
6584:+
6576:1
6562:=
6557:4
6536:(
6519:1
6505:+
6497:1
6486:5
6475:1
6461:+
6453:1
6439:=
6434:4
6394:Ο
6389:1
6384:Ο
6373:Ο
6354:k
6350:a
6345:/
6341:1
6319:4
6315:/
6311:1
6307:)
6301:4
6297:y
6290:1
6287:(
6284:=
6281:)
6278:y
6275:(
6272:f
6249:)
6244:2
6239:1
6236:+
6233:k
6229:y
6225:+
6220:1
6217:+
6214:k
6210:y
6206:+
6203:1
6200:(
6195:1
6192:+
6189:k
6185:y
6179:3
6176:+
6173:k
6170:2
6166:2
6157:4
6153:)
6147:1
6144:+
6141:k
6137:y
6133:+
6130:1
6127:(
6122:k
6118:a
6114:=
6109:1
6106:+
6103:k
6099:a
6092:,
6086:)
6083:)
6078:k
6074:y
6070:(
6067:f
6064:+
6061:1
6058:(
6054:/
6050:)
6047:)
6042:k
6038:y
6034:(
6031:f
6025:1
6022:(
6019:=
6014:1
6011:+
6008:k
6004:y
5978:2
5973:4
5967:6
5964:=
5959:0
5955:a
5948:,
5945:1
5937:2
5932:=
5927:0
5923:y
5893:Ο
5881:Ο
5855:2
5851:/
5847:3
5844:+
5841:k
5838:3
5828:3
5824:)
5820:!
5817:k
5814:(
5811:!
5808:)
5805:k
5802:3
5799:(
5794:)
5791:k
5785:+
5779:(
5776:!
5773:)
5770:k
5767:6
5764:(
5759:k
5755:)
5751:1
5745:(
5732:0
5729:=
5726:k
5715:=
5707:1
5667:k
5664:4
5654:4
5650:)
5646:!
5643:k
5640:(
5635:)
5632:k
5626:+
5620:(
5617:!
5614:)
5611:k
5608:4
5605:(
5592:0
5589:=
5586:k
5570:2
5565:2
5559:=
5551:1
5531:)
5508:3
5497:8
5494:+
5489:7
5486:1
5472:=
5449:.
5443:m
5414:)
5409:)
5404:)
5397:+
5394:1
5390:(
5384:7
5381:3
5376:+
5373:1
5369:(
5363:5
5360:2
5355:+
5352:1
5348:(
5342:3
5339:1
5334:+
5331:1
5328:=
5320:!
5317:)
5314:1
5311:+
5308:k
5305:2
5302:(
5288:2
5284:!
5280:k
5275:k
5271:2
5252:0
5249:=
5246:k
5238:=
5232:!
5229:!
5226:)
5223:1
5220:+
5217:k
5214:2
5211:(
5206:!
5203:k
5190:0
5187:=
5184:k
5176:=
5167:2
5152:+
5144:3
5140:)
5134:2
5130:x
5126:+
5123:1
5120:(
5114:5
5110:x
5101:5
5095:3
5090:4
5084:2
5078:+
5070:2
5066:)
5060:2
5056:x
5052:+
5049:1
5046:(
5040:3
5036:x
5028:3
5025:2
5020:+
5012:2
5008:x
5004:+
5001:1
4997:x
4992:=
4984:k
4980:)
4974:2
4970:x
4966:+
4963:1
4960:(
4956:!
4953:!
4950:)
4947:1
4944:+
4941:k
4938:2
4935:(
4928:k
4925:2
4921:x
4916:!
4913:!
4910:)
4907:k
4904:2
4901:(
4888:0
4885:=
4882:k
4869:2
4865:x
4861:+
4858:1
4854:x
4849:=
4842:x
4802:)
4795:+
4787:3
4783:3
4776:7
4772:1
4759:2
4755:3
4748:5
4744:1
4739:+
4731:1
4727:3
4720:3
4716:1
4703:0
4699:3
4692:1
4688:1
4682:(
4671:=
4665:1
4662:+
4659:k
4656:2
4649:k
4645:)
4639:3
4636:1
4628:(
4615:0
4612:=
4609:k
4594:=
4588:1
4585:+
4582:k
4579:2
4572:k
4565:)
4561:3
4555:(
4542:0
4539:=
4536:k
4521:=
4454:1
4451:+
4448:2
4443:+
4440:2
4435:+
4432:2
4427:+
4424:2
4419:+
4416:2
4411:+
4408:2
4403:+
4400:2
4395:+
4392:2
4384:2
4342:Ο
4309:1
4295:+
4287:1
4265:1
4254:7
4251:+
4243:1
4229:=
4224:4
4188:1
4174:+
4166:1
4155:5
4144:1
4130:+
4122:1
4108:=
4103:4
4064:1
4053:5
4042:1
4031:8
4028:+
4020:1
4006:=
4001:4
3965:1
3954:+
3946:1
3935:2
3932:+
3927:8
3924:1
3913:6
3910:=
3905:4
3887:Ο
3876:Ο
3863:y
3859:x
3852:y
3848:x
3832:.
3829:)
3826:i
3823:+
3820:1
3817:(
3812:4
3799:2
3795:2
3791:=
3788:)
3785:i
3776:(
3768:4
3764:)
3760:i
3757:+
3754:5
3751:(
3731:x
3729:(
3701:1
3685:5
3682:1
3671:4
3668:=
3663:4
3627:Ο
3623:Ο
3604:Ο
3593:Ο
3560:.
3555:7
3552:1
3547:3
3525:3
3508:p
3499:P
3492:p
3485:P
3478:p
3471:P
3464:p
3457:P
3439:.
3432:n
3429:2
3425:P
3419:n
3415:p
3409:=
3404:n
3401:2
3397:p
3391:,
3383:n
3379:P
3375:+
3370:n
3366:p
3358:n
3354:P
3348:n
3344:p
3340:2
3334:=
3329:n
3326:2
3322:P
3307:k
3300:k
3296:P
3289:k
3285:p
3279:Ο
3231:Ο
3227:Ο
3147:Ο
3118:5
3109:Ο
3104:Ο
3073:Ο
3062:Ο
3053:Ο
3046:Ο
3030:(
3002:(
2999:7
2990:Ο
2977:Ο
2970:Ο
2965:.
2959:Ο
2952:Ο
2948:Ο
2930:Ο
2919:Ο
2915:Ο
2910:.
2908:Ο
2903:.
2901:Ο
2891:Ο
2880:Ο
2869:Ο
2848:Ο
2836:Ο
2828:Ο
2799:.
2791:2
2787:/
2783:3
2780:+
2777:k
2774:3
2764:3
2760:)
2756:!
2753:k
2750:(
2747:!
2744:)
2741:k
2738:3
2735:(
2730:)
2727:k
2721:+
2715:(
2712:!
2709:)
2706:k
2703:6
2700:(
2695:k
2691:)
2687:1
2681:(
2668:0
2665:=
2662:k
2651:=
2643:1
2628:Ο
2617:Ο
2606:Ο
2602:Ο
2598:Ο
2579:Ο
2553:Ο
2518:2
2488:Ο
2484:Ο
2462:k
2459:4
2449:4
2445:)
2441:!
2438:k
2435:(
2430:)
2427:k
2421:+
2415:(
2412:!
2409:)
2406:k
2403:4
2400:(
2387:0
2384:=
2381:k
2365:2
2360:2
2354:=
2346:1
2331:Ο
2309:Ο
2304:Ο
2293:Ο
2284:Γ
2242:Ο
2216:5
2207:4
2204:=
2195:4
2192:1
2139:)
2132:9
2129:8
2119:7
2116:8
2109:(
2099:)
2092:7
2089:6
2079:5
2076:6
2069:(
2059:)
2052:5
2049:4
2039:3
2036:4
2029:(
2019:)
2012:3
2009:2
1999:1
1996:2
1989:(
1984:=
1980:)
1973:1
1970:+
1967:n
1964:2
1959:n
1956:2
1944:1
1938:n
1935:2
1930:n
1927:2
1920:(
1909:1
1906:=
1903:n
1895:=
1889:1
1881:2
1877:n
1873:4
1866:2
1862:n
1858:4
1845:1
1842:=
1839:n
1831:=
1826:2
1794:Ο
1771:Ο
1765:(
1752:Ο
1687:,
1675:2
1658:Ο
1640:n
1636:Ο
1616:n
1613:5
1610:+
1605:3
1601:n
1597:4
1592:1
1589:+
1584:2
1580:n
1567:1
1561:n
1558:2
1551:n
1547:)
1543:1
1537:(
1525:+
1520:7
1517:1
1507:5
1504:1
1499:+
1494:3
1491:1
1483:1
1477:4
1473:/
1440:Ο
1429:n
1424:n
1422:S
1418:Ο
1389:)
1382:+
1374:3
1370:3
1363:7
1359:1
1346:2
1342:3
1335:5
1331:1
1326:+
1320:3
1314:3
1310:1
1302:1
1298:(
1287:=
1281:1
1278:+
1275:k
1272:2
1265:k
1261:)
1255:3
1252:1
1244:(
1231:0
1228:=
1225:k
1210:=
1204:1
1201:+
1198:k
1195:2
1188:k
1181:)
1177:3
1171:(
1158:0
1155:=
1152:k
1137:=
1112::
1109:)
1104:3
1098:/
1094:1
1091:(
1082:6
1079:=
1052:)
1045:+
1040:7
1037:1
1027:5
1024:1
1019:+
1014:3
1011:1
1003:1
999:(
995:4
992:=
967::
964:)
961:1
958:(
949:4
946:=
929:Ο
895:(
871:Ο
833:Ο
818:7
808:Ο
802:Ο
763:Ο
735:,
730:2
721:/
714:+
707:/
703:8
700:+
697:3
665:7
656:Ο
634:"
555:Ο
535:8
526:Ο
510:Ο
495:7
486:Ο
467:Ο
446:Ο
440:Ο
432:Ο
394:Ο
392:(
371:e
364:t
357:v
343:Ο
335:Ο
69:Ο
41:.
39:Ο
34:Ο
27:Ο
21:Ο
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