10951:
10074:
10946:{\displaystyle {\begin{array}{lllll}3.254444\ldots &=3.25{\overline {4}}&={\begin{Bmatrix}\mathbf {I} =3&\mathbf {A} =25&\mathbf {P} =4\\&\#\mathbf {A} =2&\#\mathbf {P} =1\end{Bmatrix}}&={\dfrac {3254-325}{900}}&={\dfrac {2929}{900}}\\\\0.512512\ldots &=0.{\overline {512}}&={\begin{Bmatrix}\mathbf {I} =0&\mathbf {A} =\emptyset &\mathbf {P} =512\\&\#\mathbf {A} =0&\#\mathbf {P} =3\end{Bmatrix}}&={\dfrac {512-0}{999}}&={\dfrac {512}{999}}\\\\1.09191\ldots &=1.0{\overline {91}}&={\begin{Bmatrix}\mathbf {I} =1&\mathbf {A} =0&\mathbf {P} =91\\&\#\mathbf {A} =1&\#\mathbf {P} =2\end{Bmatrix}}&={\dfrac {1091-10}{990}}&={\dfrac {1081}{990}}\\\\1.333\ldots &=1.{\overline {3}}&={\begin{Bmatrix}\mathbf {I} =1&\mathbf {A} =\emptyset &\mathbf {P} =3\\&\#\mathbf {A} =0&\#\mathbf {P} =1\end{Bmatrix}}&={\dfrac {13-1}{9}}&={\dfrac {12}{9}}&={\dfrac {4}{3}}\\\\0.3789789\ldots &=0.3{\overline {789}}&={\begin{Bmatrix}\mathbf {I} =0&\mathbf {A} =3&\mathbf {P} =789\\&\#\mathbf {A} =1&\#\mathbf {P} =3\end{Bmatrix}}&={\dfrac {3789-3}{9990}}&={\dfrac {3786}{9990}}&={\dfrac {631}{1665}}\end{array}}}
11602:
9081:
9938:
496:
73:
11324:
32:
8555:
8692:
175:
11597:{\displaystyle {\begin{aligned}0.{\overline {142857}}&={\frac {142857}{10^{6}}}+{\frac {142857}{10^{12}}}+{\frac {142857}{10^{18}}}+\cdots =\sum _{n=1}^{\infty }{\frac {142857}{10^{6n}}}\\\implies &\quad {\frac {a}{1-r}}={\frac {\frac {142857}{10^{6}}}{1-{\frac {1}{10^{6}}}}}={\frac {142857}{10^{6}-1}}={\frac {142857}{999999}}={\frac {1}{7}}\end{aligned}}}
8317:
13485:
8348:
13667:
1480:: Informally, repeating decimals are often represented by an ellipsis (three periods, 0.333...), especially when the previous notational conventions are first taught in school. This notation introduces uncertainty as to which digits should be repeated and even whether repetition is occurring at all, since such ellipses are also employed for
8677:
9076:{\displaystyle {\begin{aligned}x&=0.{\overline {a_{1}a_{2}\cdots a_{n}}}\\10^{n}x&=a_{1}a_{2}\cdots a_{n}.{\overline {a_{1}a_{2}\cdots a_{n}}}\\\left(10^{n}-1\right)x=99\cdots 99x&=a_{1}a_{2}\cdots a_{n}\\x&={\frac {a_{1}a_{2}\cdots a_{n}}{10^{n}-1}}={\frac {a_{1}a_{2}\cdots a_{n}}{99\cdots 99}}\end{aligned}}}
12153:
4294:
0, 0, 1, 0, 0, 1, 6, 0, 1, 0, 2, 1, 6, 6, 1, 0, 16, 1, 18, 0, 6, 2, 22, 1, 0, 6, 3, 6, 28, 1, 15, 0, 2, 16, 6, 1, 3, 18, 6, 0, 5, 6, 21, 2, 1, 22, 46, 1, 42, 0, 16, 6, 13, 3, 2, 6, 18, 28, 58, 1, 60, 15, 6, 0, 6, 2, 33, 16, 22, 6, 35, 1, 8, 3, 1, 18, 6, 6, 13, 0, 9, 5, 41, 6, 16, 21, 28, 2, 44, 1, 6,
1614:
For any given divisor, only finitely many different remainders can occur. In the example above, the 74 possible remainders are 0, 1, 2, ..., 73. If at any point in the division the remainder is 0, the expansion terminates at that point. Then the length of the repetend, also called
4482:
3, 11, 37, 101, 41, 7, 239, 73, 333667, 9091, 21649, 9901, 53, 909091, 31, 17, 2071723, 19, 1111111111111111111, 3541, 43, 23, 11111111111111111111111, 99990001, 21401, 859, 757, 29, 3191, 211, 2791, 353, 67, 103, 71, 999999000001, 2028119, 909090909090909091, 900900900900990990990991, 1676321, 83,
5296:
The rotation of the repetend of a cyclic number always happens in such a way that each successive repetend is a bigger number than the previous one. In the succession above, for instance, we see that 0.142857... < 0.285714... < 0.428571... < 0.571428... < 0.714285... < 0.857142....
1618:
If 0 never occurs as a remainder, then the division process continues forever, and eventually, a remainder must occur that has occurred before. The next step in the division will yield the same new digit in the quotient, and the same new remainder, as the previous time the remainder was the same.
1503:
may be read "eleven point repeating one double eight six seven nine two four five two eight three zero", "eleven point repeated one double eight six seven nine two four five two eight three zero", "eleven point recurring one double eight six seven nine two four five two eight three zero" "eleven
13493:
8147:
13219:
1560:
etc. Observe that at each step we have a remainder; the successive remainders displayed above are 56, 42, 50. When we arrive at 50 as the remainder, and bring down the "0", we find ourselves dividing 500 by 74, which is the same problem we began with. Therefore, the decimal repeats:
4433:
0, 1, 0, 6, 2, 6, 16, 18, 22, 28, 15, 3, 5, 21, 46, 13, 58, 60, 33, 35, 8, 13, 41, 44, 96, 4, 34, 53, 108, 112, 42, 130, 8, 46, 148, 75, 78, 81, 166, 43, 178, 180, 95, 192, 98, 99, 30, 222, 113, 228, 232, 7, 30, 50, 256, 262, 268, 5, 69, 28, 141, 146, 153, 155, 312, 79... (sequence
8550:{\displaystyle {\begin{aligned}7.48181818\ldots &=7.3+0.18181818\ldots \\&={\frac {73}{10}}+{\frac {18}{99}}={\frac {73}{10}}+{\frac {9\cdot 2}{9\cdot 11}}={\frac {73}{10}}+{\frac {2}{11}}\\&={\frac {11\cdot 73+10\cdot 2}{10\cdot 11}}={\frac {823}{110}}\end{aligned}}}
2418:
11157:
9292:
Any repeating decimal not of the form described above can be written as a sum of a terminating decimal and a repeating decimal of one of the two above types (actually the first type suffices, but that could require the terminating decimal to be negative). For example,
8566:
4388:
0, 0, 3, 0, 0, 6, 142857, 0, 1, 0, 09, 3, 076923, 714285, 6, 0, 0588235294117647, 5, 052631578947368421, 0, 047619, 45, 0434782608695652173913, 6, 0, 384615, 037, 571428, 0344827586206896551724137931, 3, 032258064516129, 0, 03, 2941176470588235, 285714... (sequence
14603:
Repeating decimals (also called decimal sequences) have found cryptographic and error-correction coding applications. In these applications repeating decimals to base 2 are generally used which gives rise to binary sequences. The maximum length binary sequence for
12357:
14391:
12252:
11313:
14946:
2523:
7073:
An integer that is not coprime to 10 but has a prime factor other than 2 or 5 has a reciprocal that is eventually periodic, but with a non-repeating sequence of digits that precede the repeating part. The reciprocal can be expressed as:
12046:
8116:
15052:
4545:
7, 3, 103, 53, 11, 79, 211, 41, 73, 281, 353, 37, 2393, 449, 3061, 1889, 137, 2467, 16189, 641, 3109, 4973, 11087, 1321, 101, 7151, 7669, 757, 38629, 1231, 49663, 12289, 859, 239, 27581, 9613, 18131, 13757, 33931... (sequence
195:
The article is polluted with multiple non-encyclopedic tables, often placed before the true content that is hidden by them. Section are placed in a random order with the most important facts relegated toward the end of the
6985:
8312:{\displaystyle {\begin{aligned}x&=0.000000100000010000001\ldots \\10^{7}x&=1.000000100000010000001\ldots \\\left(10^{7}-1\right)x=9999999x&=1\\x&={\frac {1}{10^{7}-1}}={\frac {1}{9999999}}\end{aligned}}}
13480:{\displaystyle {\frac {1}{k}}={\frac {1}{b}}+{\frac {(b-k)^{1}}{b^{2}}}+{\frac {(b-k)^{2}}{b^{3}}}+{\frac {(b-k)^{3}}{b^{4}}}+\cdots +{\frac {(b-k)^{N-1}}{b^{N}}}+\cdots ={\frac {1}{b}}{\frac {1}{1-{\frac {b-k}{b}}}}.}
12986:
1498:
may be read "one point two repeating three four", "one point two repeated three four", "one point two recurring three four", "one point two repetend three four" or "one point two into infinity three four". Likewise,
13082:
14555:
8697:
7151:
14474:
8353:
7334:
7244:
13662:{\displaystyle {\frac {1}{7}}=\left({\frac {1}{10^{\phantom {1}}}}+{\frac {5}{10^{2}}}+{\frac {21}{10^{3}}}+{\frac {A5}{10^{4}}}+{\frac {441}{10^{5}}}+{\frac {1985}{10^{6}}}+\cdots \right)_{\text{base 12}}}
2257:
269:(that is, after some place, the same sequence of digits is repeated forever); if this sequence consists only of zeros (that is if there is only a finite number of nonzero digits), the decimal is said to be
11046:
11329:
8571:
7411:
12550:
8672:{\displaystyle {\begin{aligned}11.18867924528301886792452830\ldots &=11+0.18867924528301886792452830\ldots \\&=11+{\frac {10}{53}}={\frac {11\cdot 53+10}{53}}={\frac {593}{53}}\end{aligned}}}
6887:
8152:
5850:
2600:
12633:
14689:
12267:
7767:
14297:
1891:
5297:
This, for cyclic fractions with long repetends, allows us to easily predict what the result of multiplying the fraction by any natural number n will be, as long as the repetend is known.
12168:
7843:
7908:
15132:
6412:
2702:
1619:
Therefore, the following division will repeat the same results. The repeating sequence of digits is called "repetend" which has a certain length greater than 0, also called "period".
1504:
point repetend one double eight six seven nine two four five two eight three zero" or "eleven point into infinity one double eight six seven nine two four five two eight three zero".
2213:
1997:
12468:
A rational number has a terminating sequence if all the prime factors of the denominator of the fully reduced fractional form are also factors of the base. These numbers make up a
7964:
1784:
11216:
2148:
2079:
11026:
7644:
7591:
10066:
will be represented by zero, which being to the left of the other digits, will not affect the final result, and may be omitted in the calculation of the generating function.
14858:
10033:
9988:
9932:
9885:
2120:
351:
rather than a repeating decimal, since the zeros can be omitted and the decimal terminates before these zeros. Every terminating decimal representation can be written as a
10995:
10064:
9907:
9860:
9838:
9732:
digits after the decimal point that do not belong to the repeating part, can be written as a (not necessarily reduced) fraction whose denominator is (10 − 1)10.
10973:
5335:
61, 131, 181, 461, 491, 541, 571, 701, 811, 821, 941, 971, 1021, 1051, 1091, 1171, 1181, 1291, 1301, 1349, 1381, 1531, 1571, 1621, 1741, 1811, 1829, 1861,... (sequence
2425:
2051:
1938:
7937:
2242:
8017:
7993:
4350:
0, 0, 2, 0, 4, 2, 3, 0, 6, 4, 10, 2, 12, 3, 4, 0, 8, 6, 18, 4, 6, 10, 11, 2, 20, 12, 18, 3, 28, 4, 5, 0, 10, 8, 12, 6, 36, 18, 12, 4, 20, 6, 14, 10, 12, 11, ... (=
7869:
7617:
2648:
2025:
7697:
7673:
12148:{\displaystyle {\text{period}}\left({\frac {1}{p}}\right)={\text{period}}\left({\frac {1}{p^{2}}}\right)=\cdots ={\text{period}}\left({\frac {1}{p^{m}}}\right)}
10008:
9963:
8043:
7804:
7723:
7564:
2722:
2622:
1911:
1844:
1824:
1804:
423:
number with a terminating decimal representation also trivially has a second, alternative representation as a repeating decimal whose repetend is the digit
12903:. The repeating sequence is preceded by a transient of finite length if the reduced fraction also shares a prime factor with the base. A repeating sequence
8049:
14965:
9093:
digits long, first occurring right after the decimal point, then the fraction (not necessarily reduced) will be the integer number represented by the
8342:
9s. Knowing just that, a general repeating decimal can be expressed as a fraction without having to solve an equation. For example, one could reason:
6898:
5312: − 1. In such primes, each digit 0, 1,..., 9 appears in the repeating sequence the same number of times as does each other digit (namely,
12912:
336:, which becomes periodic after the decimal point, repeating the 13-digit pattern "1886792452830" forever, i.e. 11.18867924528301886792452830....
13090:
An irrational number has a representation of infinite length that is not, from any point, an indefinitely repeating sequence of finite length.
12997:
15140:
7080:
15102:
2413:{\displaystyle x=y+\sum _{n=1}^{\infty }{\frac {c}{{(10^{k})}^{n}}}=y+\left(c\sum _{n=0}^{\infty }{\frac {1}{{(10^{k})}^{n}}}\right)-c.}
7263:
7173:
14485:
11152:{\displaystyle 0.{\overline {1}}={\frac {1}{10}}+{\frac {1}{100}}+{\frac {1}{1000}}+\cdots =\sum _{n=1}^{\infty }{\frac {1}{10^{n}}}}
6533:
5741:
5409:
5342:
5061:
4553:
4490:
4441:
4396:
4302:
427:. This is obtained by decreasing the final (rightmost) non-zero digit by one and appending a repetend of 9. Two examples of this are
14402:
11040:. That is, a repeating decimal can be regarded as the sum of an infinite number of rational numbers. To take the simplest example,
7353:
15078:
14818:
12500:
320:
digit following the decimal point and then repeats the sequence "144" forever, i.e. 5.8144144144.... Another example of this is
137:
12684:
109:
12442:
Various features of repeating decimals extend to the representation of numbers in all other integer bases, not just base 10:
9717:(denominator has three 9s and one 0 because three digits repeat and there is one non-repeating digit after the decimal point)
6833:
9682:(denominator has one 9 and two 0s because one digit repeats and there are two non-repeating digits after the decimal point)
5858:
Those reciprocals of primes can be associated with several sequences of repeating decimals. For example, the multiples of
5801:
116:
2530:
12561:
12352:{\displaystyle {\text{period}}\left({\frac {1}{p^{m+c}}}\right)=p^{c}\cdot {\text{period}}\left({\frac {1}{p}}\right).}
517:
200:
90:
45:
14632:
14386:{\displaystyle {\frac {bp}{q}}-1\;\;<\;\;z=\left\lfloor {\frac {bp}{q}}\right\rfloor \;\;\leq \;\;{\frac {bp}{q}},}
456:. Their decimal representation neither terminates nor infinitely repeats, but extends forever without repetition (see
556:
There are several notational conventions for representing repeating decimals. None of them are accepted universally.
543:
236:
218:
156:
59:
14850:
12247:{\displaystyle {\text{period}}\left({\frac {1}{p^{m}}}\right)\neq {\text{period}}\left({\frac {1}{p^{m+1}}}\right),}
7729:
5747:
The reason is that 3 is a divisor of 9, 11 is a divisor of 99, 41 is a divisor of 99999, etc. To find the period of
4583:
to 10) always produces a repeating decimal. The length of the repetend (period of the repeating decimal segment) of
525:
15263:
1849:
1626:, its denominator has any prime factors besides 2 or 5, or in other words, cannot be expressed as 2 5, where
123:
12794:
to the same real number – and there are no other duplicate images. In the decimal system, for example, there is 0.
280:
if and only if its decimal representation is repeating or terminating. For example, the decimal representation of
11618:
The cyclic behavior of repeating decimals in multiplication also leads to the construction of integers which are
7810:
7875:
15365:
521:
94:
6368:
105:
2653:
1391:
15221:
William E. Heal. Some
Properties of Repetends. Annals of Mathematics, Vol. 3, No. 4 (Aug., 1887), pp. 97–103
15197:
For primes greater than 5, all the digital roots appear to have the same value, 9. We can confirm this if...
11308:{\displaystyle {\frac {a}{1-r}}={\frac {\frac {1}{10}}{1-{\frac {1}{10}}}}={\frac {1}{10-1}}={\frac {1}{9}}}
2155:
1945:
437:. (This type of repeating decimal can be obtained by long division if one uses a modified form of the usual
1444:, the convention is to enclose the repetend in parentheses. This can cause confusion with the notation for
7943:
5253:
A fraction which is cyclic thus has a recurring decimal of even length that divides into two sequences in
1747:
1474:, the arc notation over the repetend is also used as an alternative to the vinculum and the dots notation.
15057:
14285:
9214:
digits 9 of the denominator (and, as before, the fraction may subsequently be simplified). For example,
14941:{\displaystyle \operatorname {ord} _{n}(b):=\min\{L\in \mathbb {N} \,\mid \,b^{L}\equiv 1{\bmod {n}}\}}
11198:. Because the absolute value of the common factor is less than 1, we can say that the geometric series
5402:
7, 23, 47, 59, 167, 179, 263, 383, 503, 863, 887, 983, 1019, 1367, 1487, 1619, 1823, 2063... (sequence
4626:
2125:
2056:
11000:
7623:
7570:
12839:, if the reduced fraction's denominator contains a prime factor that is not a factor of the base. If
190:
10013:
9968:
9912:
9865:
9840:
represents the digits of the integer part of the decimal number (to the left of the decimal point),
2084:
15370:
7539:
Given a repeating decimal, it is possible to calculate the fraction that produces it. For example:
506:
10978:
10047:
9890:
9843:
9821:
5071:
multiple of a cyclic number (that is, a multiple having the same number of digits) is a rotation:
2518:{\displaystyle \textstyle \sum _{n=0}^{\infty }{\frac {1}{{(10^{k})}^{n}}}={\frac {1}{1-10^{-k}}}}
6189:
4610:
510:
83:
10958:
5308:
which ends in the digit 1 in base 10 and whose reciprocal in base 10 has a repetend with length
4888:
010309278350515463917525773195876288659793814432989690721649484536082474226804123711340206185567
14730:
5226:
The reason for the cyclic behavior is apparent from an arithmetic exercise of long division of
1646:
with integer coefficients, and its unique solution is a rational number. In the example above,
258:
130:
51:
15110:
14568:, there can be only a finite number of them with the consequence that they must recur in the
10079:
6643:
6627:
5987:
where the repetend of each fraction is a cyclic re-arrangement of 076923. The second set is:
5395:
4566:
2030:
14698: − 1 have an autocorrelation function that has a negative peak of −1 for shift of
4321:
1916:
15243:
Mitchell, Douglas W., "A nonlinear random number generator with known, long cycle length",
15163:
14766:
12859:
11860:
11613:
7919:
5254:
4602:
2220:
1445:
7999:
7975:
7439:) digits after the decimal point. Some or all of the digits in the transient can be zeros.
8:
12886:
12488:
8686:-digit period (repetend length), beginning right after the decimal point, as a fraction:
7851:
7599:
6343:
6339:
6260:
2627:
2004:
13182:
repeats with period length 4, in contrast with the equivalent decimal expansion of 0.2;
12679:
repeating part consisting of the digit 0. If the base is positive, then there exists an
11202:
and find the exact value in the form of a fraction by using the following formula where
7679:
7655:
15317:
15188:
15180:
14951:
14735:
14592:
14277:
12688:
11619:
9993:
9948:
8028:
7789:
7708:
7549:
6486:. There are three known primes for which this is not true, and for those the period of
5365:
5354:
5350:
2707:
2607:
1896:
1829:
1809:
1789:
457:
438:
300:, repeating the single digit "3" forever, i.e. 0.333.... A more complicated example is
20:
15341:
14799:
Beswick, Kim (2004), "Why Does 0.999... = 1?: A Perennial
Question and Number Sense",
15338:
15192:
14740:
14573:
12680:
11199:
9722:
8111:{\displaystyle ={\frac {828}{990}}={\frac {18\cdot 46}{18\cdot 55}}={\frac {46}{55}}}
1609:
1494:
In
English, there are various ways to read repeating decimals aloud. For example, 1.2
1481:
568:
453:
266:
185:
15047:{\displaystyle \lambda (n):=\max\{\operatorname {ord} _{n}(b)\,\mid \,\gcd(b,n)=1\}}
13755:) there is the following algorithm producing the repetend together with its length:
11666:
Various properties of repetend lengths (periods) are given by
Mitchell and Dickson.
15172:
14745:
12803:
11163:
6174:
461:
352:
9818:
The following picture suggests kind of compression of the above shortcut. Thereby
7968:(collate 2nd repetition here with 1st above = move by 2 places = multiply by 100)
12455:
11037:
9646:
An even faster method is to ignore the decimal point completely and go like this
4572:
1643:
1623:
1513:
277:
262:
12675:, then a terminating sequence is obviously equivalent to the same sequence with
12843:
is the maximal factor of the reduced denominator which is coprime to the base,
8682:
It is possible to get a general formula expressing a repeating decimal with an
6980:{\displaystyle \operatorname {LCM} (T_{p^{k}},T_{q^{\ell }},T_{r^{m}},\ldots )}
6213:
1394:, the convention is to place dots above the outermost numerals of the repetend.
1359:
1305:
1637:
1324:, the convention is to draw a horizontal line (a vinculum) above the repetend.
15359:
15258:
14750:
14718:
12451:
8137:
digits, all of which are 0 except the final one which is 1. For instance for
4686:
4673: − 1 then the repetend, expressed as an integer, is called a
4227:
1517:
1459:
1277:
297:
15161:
Gray, Alexander J. (March 2000). "Digital roots and reciprocals of primes".
9159:
since the repeating block is 012 (a 3-digit block); this further reduces to
14755:
12981:{\displaystyle \left(0.{\overline {A_{1}A_{2}\ldots A_{\ell }}}\right)_{b}}
4644:
of the repetend of the reciprocal of any prime number greater than 5 is 9.
4641:
4576:
1371:
356:
14787:
What Is
Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed.
12835:
A rational number has an indefinitely repeating sequence of finite length
6526:
divides 10−1. These three primes are 3, 487, and 56598313 (sequence
6100:
where the repetend of each fraction is a cyclic re-arrangement of 153846.
5874:
can be divided into two sets, with different repetends. The first set is:
15245:
7912:(move decimal to start of repetition = move by 1 place = multiply by 10)
6579:
are primes other than 2 or 5, the decimal representation of the fraction
6275:
is a prime other than 2 or 5, the decimal representation of the fraction
1417:
1383:
1363:
1301:
576:
458:§ Every rational number is either a terminating or repeating decimal
15277:
Armstrong, N. J., and
Armstrong, R. J., "Some properties of repetends",
13077:{\displaystyle {\frac {(A_{1}A_{2}\ldots A_{\ell })_{b}}{b^{\ell }-1}}.}
9089:
If the repeating decimal is between 0 and 1, and the repeating block is
15184:
13095:
9794:= 6, because 999999 = 7 × 142857. The period of the fraction
5369:
14591:
of the repetend equals the number of the remainders (see also section
12446:
Every real number can be represented as an integer part followed by a
4621: − 1; if not, then the repetend length is a factor of
15346:
12469:
9234:
since the repeating block is 4 and this block is preceded by 3 zeros,
8133:
The procedure below can be applied in particular if the repetend has
7146:{\displaystyle {\frac {1}{2^{a}\cdot 5^{b}p^{k}q^{\ell }\cdots }}\,,}
6255: − 1 and differ only by a cyclic permutation. Such numbers
1463:
1387:
1367:
347:. If the repetend is a zero, this decimal representation is called a
15176:
5769:
divides some number 999...999 in which the number of digits divides
5421:
Some reciprocals of primes that do not generate cyclic numbers are:
495:
72:
15316:
Bellamy, J. "Randomness of D sequences via diehard testing". 2013.
15303:
Kak, Subhash, "Encryption and error-correction using d-sequences".
14760:
12692:
12454:
to non-decimal systems) followed by a finite or infinite number of
11976:
have no common prime factors other than 2 or 5, then the period of
1732:
The process of how to find these integer coefficients is described
1379:
1339:
1331:
1313:
1309:
584:
428:
15321:
14593:
Every rational number is either a terminating or repeating decimal
6266:
6103:
In general, the set of proper multiples of reciprocals of a prime
1610:
Every rational number is either a terminating or repeating decimal
15133:"Prove that every repeating decimal represents a rational number"
11781:
9937:
8322:
So this particular repeating decimal corresponds to the fraction
7442:
A subsequent repetend which is the same as that for the fraction
7329:{\displaystyle {\frac {2^{b-a}}{10^{b}p^{k}q^{\ell }\cdots }}\,,}
7239:{\displaystyle {\frac {5^{a-b}}{10^{a}p^{k}q^{\ell }\cdots }}\,,}
6422:
5247:
4580:
1437:
1433:
1413:
1409:
1405:
1351:
1343:
1335:
1293:
449:
14550:{\displaystyle zq\leq bp\quad \implies \quad 0\leq bp-zq=:p'\,.}
11825:
is not divisible by 2 or 5, then the length of the transient of
12700:
into some closed interval of the reals, which maps the strings
12407:, then each digit 0, 1, ..., 9 appears in the repetend exactly
9288:
since the repeating block is 012 and it is preceded by 2 zeros.
1733:
1471:
1467:
1441:
1429:
1425:
1421:
1401:
1321:
1289:
1281:
9253:
since the repeating block is 56 and it is preceded by 2 zeros,
1622:
In base 10, a fraction has a repeating decimal if and only if
14926:
14672:
14662:
14469:{\displaystyle bp-q<zq\quad \implies \quad p':=bp-zq<q}
12447:
9909:
being the string of repeated digits (the period) with length
9735:
Conversely the period of the repeating decimal of a fraction
9202:(extra) digits 0 between the decimal point and the repeating
7534:
5855:
and then by inspection find the repetend 09 and period of 2.
1455:
1375:
1355:
1317:
1297:
1285:
445:
379:
19:"Repeating fraction" redirects here. Not to be confused with
10975:
in the examples above denotes the absence of digits of part
9198:
If the repeating decimal is as above, except that there are
7068:
4866:
016393442622950819672131147540983606557377049180327868852459
1507:
16:
Decimal representation of a number whose digits are periodic
7406:{\displaystyle {\frac {1}{10^{a}p^{k}q^{\ell }\cdots }}\,,}
7011:,... are respectively the period of the repeating decimals
6528:
5736:
5404:
5337:
5056:
4548:
4485:
4436:
4391:
4351:
4297:
1638:
Every repeating or terminating decimal is a rational number
1347:
15336:
15268:, Chelsea Publ. Co., 1952 (orig. 1918), pp. 164–173.
12545:{\displaystyle b\in \mathbb {Z} \smallsetminus \{-1,0,1\}}
9194:= 1, since the repeating block is 9 (also a 1-digit block)
4844:
0169491525423728813559322033898305084745762711864406779661
14717:. The randomness of these sequences has been examined by
7514:
there are 6 repeating digits, 571428, the same amount as
1516:
represented as a fraction into decimal form, one may use
13202:
has period 6 in duodecimal, just as it does in decimal.
11607:
11031:
11661:
11028:
and a corresponding absence in the generated fraction.
9561:
or alternatively 0.3789789... = −0.6 + 0.9789789... = −
6882:{\displaystyle {\frac {1}{p^{k}q^{\ell }r^{m}\cdots }}}
5773: − 1. Since the period is never greater than
1485:
472:
14789:
Oxford, England: Oxford
University Press, 1996: p. 67.
10797:
10614:
10448:
10282:
10116:
4625: − 1. This result can be deduced from
2656:
2429:
15290:
Kak, Subhash, Chatterjee, A. "On decimal sequences".
14968:
14861:
14819:"Lambert's Original Proof that $ \pi$ is irrational"
14635:
14488:
14405:
14300:
13496:
13222:
13000:
12915:
12564:
12503:
12270:
12171:
12049:
11327:
11219:
11049:
11003:
10981:
10961:
10928:
10911:
10886:
10745:
10728:
10703:
10562:
10537:
10396:
10371:
10230:
10205:
10077:
10050:
10016:
9996:
9971:
9951:
9915:
9893:
9868:
9846:
9824:
8695:
8569:
8351:
8150:
8052:
8031:
8002:
7978:
7946:
7922:
7878:
7854:
7813:
7792:
7732:
7711:
7682:
7658:
7626:
7602:
7573:
7552:
7356:
7266:
7176:
7083:
6901:
6836:
6371:
5804:
2710:
2630:
2610:
2533:
2428:
2260:
2223:
2158:
2128:
2087:
2059:
2033:
2007:
1948:
1919:
1899:
1852:
1832:
1812:
1792:
1750:
339:
The infinitely repeated digit sequence is called the
15079:"Rational numbers have repeating decimal expansions"
11911:
equals the lowest common multiple of the periods of
11863:
of 10 mod n, that is the smallest integer such that
5242:: the sequential remainders are the cyclic sequence
4560:
1594:, the remainder at step k, for any positive integer
11994:equals the least common multiple of the periods of
9862:makes up the string of digits of the preperiod and
5845:{\displaystyle {\frac {10^{11-1}-1}{11}}=909090909}
4691:Examples of fractions belonging to this group are:
97:. Unsourced material may be challenged and removed.
15046:
14940:
14683:
14549:
14468:
14385:
13661:
13479:
13076:
12980:
12627:
12544:
12434:For some other properties of repetends, see also.
12351:
12246:
12147:
11596:
11307:
11151:
11020:
10989:
10967:
10945:
10058:
10027:
10002:
9982:
9957:
9926:
9901:
9879:
9854:
9832:
9379:or alternatively 1.23444... = 0.79 + 0.44444... =
9140:since the repeating block is 56 (a 2-digit block),
9075:
8671:
8549:
8311:
8110:
8037:
8011:
7987:
7958:
7931:
7902:
7863:
7837:
7798:
7761:
7717:
7691:
7667:
7638:
7611:
7585:
7558:
7405:
7328:
7238:
7145:
6979:
6881:
6406:
5844:
5777: − 1, we can obtain this by calculating
2716:
2696:
2642:
2616:
2595:{\displaystyle x=y-c+{\frac {10^{k}c}{10^{k}-1}}.}
2594:
2517:
2412:
2236:
2207:
2142:
2114:
2073:
2045:
2019:
1991:
1932:
1905:
1885:
1838:
1818:
1798:
1778:
14272:The subsequent line calculates the new remainder
14146:// mark the digits of the repetend by a vinculum:
12628:{\displaystyle D:=\{d_{1},d_{1}+1,\dots ,d_{r}\}}
11622:when multiplied by certain numbers. For example,
9121:since the repeating block is 4 (a 1-digit block),
8338:, where the denominator is the number written as
7511:there are 2 initial non-repeating digits, 03; and
5349:A prime is a proper prime if and only if it is a
15357:
15017:
14984:
14887:
14684:{\displaystyle a(i)=2^{i}{\bmod {p}}{\bmod {2}}}
14265:The first highlighted line calculates the digit
12465:sequence obviously represents a rational number.
11036:A repeating decimal can also be expressed as an
6700:008403361344537815126050420168067226890756302521
1940:separates the repeating and terminating groups:
1490:, for example, can be represented as 3.14159....
14598:
13713:
9097:-digit block divided by the one represented by
9086:More explicitly, one gets the following cases:
6267:Reciprocals of composite integers coprime to 10
9790:that makes 10 − 1 divisible by 7 is
7762:{\displaystyle ={\frac {3}{9}}={\frac {1}{3}}}
5416:
4822:0212765957446808510638297872340425531914893617
4209:0212765957446808510638297872340425531914893617
13871:// the position of the place with remainder p
11886:,... are distinct primes, then the period of
7648:(multiply each side of the above line by 10)
6219:is a prime and 10 is a primitive root modulo
4295:22, 15, 46, 18, 1, 96, 42, 2, 0... (sequence
1886:{\displaystyle n=\lceil {\log _{10}b}\rceil }
15041:
14987:
14935:
14890:
14143:// the length of the repetend (being < q)
12622:
12571:
12539:
12518:
5728:01123595505617977528089887640449438202247191
5444:, which has a period (repetend length) of 1.
5273:starts '142' and is followed by '857' while
4894:The list can go on to include the fractions
1880:
1859:
1520:. For example, consider the rational number
14572:loop. Such a recurrence is detected by the
12437:
12373:ending in a 1, that is, if the repetend of
9759:such that 10 − 1 is divisible by
9721:It follows that any repeating decimal with
7838:{\displaystyle =\ \ \ \ 0.836363636\ldots }
5250:for more properties of this cyclic number.
524:. Unsourced material may be challenged and
460:). Examples of such irrational numbers are
60:Learn how and when to remove these messages
15294:, vol. IT-27, pp. 647–652, September 1981.
14509:
14505:
14432:
14428:
14364:
14363:
14359:
14358:
14328:
14327:
14323:
14322:
13835:// all places are right to the radix point
12556:combined with a consecutive set of digits
11467:
11463:
7903:{\displaystyle =\ \ \ \ 8.36363636\ldots }
7535:Converting repeating decimals to fractions
6316:020408163265306122448979591836734693877551
1642:Each repeating decimal number satisfies a
15212:, Volume 1, Chelsea Publishing Co., 1952.
15016:
15012:
14908:
14904:
14900:
14543:
12511:
7399:
7322:
7232:
7139:
7069:Reciprocals of integers not coprime to 10
6815:, etc. are primes other than 2 or 5, and
5706:01204819277108433734939759036144578313253
4254:is the length of the (decimal) repetend.
2185:
2136:
2067:
1508:Decimal expansion and recurrence sequence
544:Learn how and when to remove this message
444:Any number that cannot be expressed as a
237:Learn how and when to remove this message
219:Learn how and when to remove this message
157:Learn how and when to remove this message
14960:) which divides the Carmichael function
9936:
7167:This fraction can also be expressed as:
6892:is a repeating decimal with a period of
6679:) and it happens to be 48 in this case:
6407:{\displaystyle a^{m}\equiv 1{\pmod {n}}}
6111:subsets, each with repetend length
2697:{\textstyle {\frac {10^{k}c}{10^{k}-1}}}
2150:is a terminating group of digits. Then,
316:, whose decimal becomes periodic at the
15292:IEEE Transactions on Information Theory
14798:
13920:// index z of digit within: 0 ≤ z ≤ b-1
5289:(by rotation) starts '857' followed by
4617:, then the repetend length is equal to
15358:
12491:is a standard one, that is it has base
9755:will be (at most) the smallest number
9206:-digit block, then one can simply add
2208:{\displaystyle c=d_{1}d_{2}\,...d_{k}}
1992:{\displaystyle 10^{n}x=ab.{\bar {c}}.}
273:, and is not considered as repeating.
15337:
14587:is the only non-constant. The length
11608:Multiplication and cyclic permutation
11032:Repeating decimals as infinite series
9945:In the generated fraction, the digit
9813:
7701:(subtract the 1st line from the 2nd)
1330:: In some Islamic countries, such as
15232:Recreations in the Theory of Numbers
15160:
15130:
15076:
14583:is formed in the yellow line, where
14564:are non-negative integers less than
14104:// append the character of the digit
11662:Other properties of repetend lengths
11206:is the first term of the series and
9463:0.3789789... = 0.3 + 0.0789789... =
7959:{\displaystyle =836.36363636\ldots }
4579:denominator other than 2 or 5 (i.e.
1779:{\displaystyle x=a.b{\overline {c}}}
522:adding citations to reliable sources
489:
355:, a fraction whose denominator is a
168:
95:adding citations to reliable sources
66:
25:
12847:is the smallest exponent such that
6827:, etc. are positive integers, then
6396:
5640:01408450704225352112676058338028169
13:
15249:17, January 1993, pp. 55–62.
14763:, a repeating decimal equal to one
11434:
11127:
11004:
10962:
10858:
10842:
10675:
10659:
10638:
10509:
10493:
10343:
10327:
10306:
10177:
10161:
10017:
9972:
9916:
9869:
6223:. Then, the decimal expansions of
2727:
2446:
2359:
2289:
560:Different notations with examples
14:
15382:
15330:
13802:// up to the digit with value b–1
9297:1.23444... = 1.23 + 0.00444... =
5765:, we can check whether the prime
5618:014925373134328358208955223880597
4680:
4561:Fractions with prime denominators
2143:{\displaystyle y\in \mathbb {Z} }
2074:{\displaystyle k\in \mathbb {N} }
1671:= 58144.144144... − 58.144144...
276:It can be shown that a number is
41:This article has multiple issues.
15264:History of the Theory of Numbers
15210:History of the Theory of Numbers
15103:"The Sets of Repeating Decimals"
11021:{\displaystyle \#\mathbf {A} =0}
11008:
10983:
10862:
10846:
10827:
10814:
10801:
10679:
10663:
10644:
10631:
10618:
10513:
10497:
10478:
10465:
10452:
10347:
10331:
10312:
10299:
10286:
10181:
10165:
10146:
10133:
10120:
10052:
10021:
9976:
9920:
9895:
9873:
9848:
9826:
7639:{\displaystyle =3.333333\ldots }
7586:{\displaystyle =0.333333\ldots }
6251: − 1, all have period
4403:The decimal repetend lengths of
494:
296:becomes periodic just after the
173:
71:
30:
15310:
15307:, vol. C-34, pp. 803–809, 1985.
15297:
15284:
15281:87, November 2003, pp. 437–443.
15271:
15252:
15237:
15224:
15215:
14622:(when 2 is a primitive root of
14510:
14504:
14433:
14427:
13490:For example 1/7 in duodecimal:
12450:point (the generalization of a
11470:
10038:Note that in the absence of an
6389:
6198:In particular, it follows that
6126:
1739:
378:); it may also be written as a
82:needs additional citations for
49:or discuss these issues on the
15202:
15154:
15124:
15095:
15070:
15032:
15020:
15009:
15003:
14978:
14972:
14881:
14875:
14835:
14811:
14801:Australian Mathematics Teacher
14792:
14779:
14645:
14639:
14506:
14429:
13394:
13381:
13347:
13334:
13306:
13293:
13265:
13252:
13041:
13004:
12899:which in turn is smaller than
11464:
10997:in the decimal, and therefore
10028:{\displaystyle \#\mathbf {A} }
9983:{\displaystyle \#\mathbf {P} }
9927:{\displaystyle \#\mathbf {P} }
9880:{\displaystyle \#\mathbf {A} }
6974:
6908:
6400:
6390:
5664:, which has a period of eight.
5532:, which has a period of three.
4268:) of the decimal repetends of
2471:
2458:
2384:
2371:
2314:
2301:
2115:{\displaystyle x=y.{\bar {c}}}
2106:
2027:), the proof is complete. For
1980:
1615:"period", is defined to be 0.
1:
15305:IEEE Transactios on Computers
14560:Because all these remainders
12461:If the base is an integer, a
12391:is a cyclic number of length
8575:11.18867924528301886792452830
8128:
8021:(subtract to clear decimals)
6504:is the same as the period of
6322:The period (repetend length)
6143:) of the decimal repetend of
5795:. For example, for 11 we get
5554:, which has a period of five.
480:
14785:Courant, R. and Robbins, H.
14599:Applications to cryptography
13714:Algorithm for positive bases
12962:
12689:right-sided infinite strings
11732:is composite, the period of
11626:. 102564 is the repetend of
11340:
11058:
10990:{\displaystyle \mathbf {A} }
10782:
10599:
10433:
10267:
10101:
10059:{\displaystyle \mathbf {I} }
9902:{\displaystyle \mathbf {P} }
9855:{\displaystyle \mathbf {A} }
9833:{\displaystyle \mathbf {I} }
8852:
8750:
8594:0.18867924528301886792452830
7431:An initial transient of max(
5488:, which has a period of six.
5466:, which has a period of two.
4800:0344827586206896551724137931
4470:has decimal repetend length
4309:For comparison, the lengths
3681:0344827586206896551724137931
1771:
7:
14849:, in terms of group theory
14724:
11642:and 410256 the repetend of
7499:= 0, and the other factors
6350:is a positive integer then
6330:(49) = 42, where
5730:, which has a period of 44.
5708:, which has a period of 41.
5686:, which has a period of 13.
5642:, which has a period of 35.
5620:, which has a period of 33.
5598:, which has a period of 13.
5576:, which has a period of 21.
5510:, which has a period of 15.
5417:Other reciprocals of primes
4324:repetends of the fractions
2001:If the decimals terminate (
1686:
1675:
1634:are non-negative integers.
485:
193:. The specific problem is:
10:
15387:
14823:Mathematics Stack Exchange
14694:These sequences of period
14284:. As a consequence of the
12885:which is a divisor of the
11611:
10968:{\displaystyle \emptyset }
9766:For example, the fraction
6358:) is the smallest integer
4684:
4647:If the repetend length of
4564:
4358:not a power of 2 else =0).
2624:is the sum of an integer (
1893:, the number of digits of
1846:are groups of digits, let
1744:Given a repeating decimal
1392:People's Republic of China
18:
13787:// b ≥ 2; 0 < p < q
12489:positional numeral system
12395: − 1 and
11851:), and the period equals
8120:(reduce to lowest terms)
7771:(reduce to lowest terms)
6539:Similarly, the period of
6326:(49) must be a factor of
6177:. The length is equal to
6131:For an arbitrary integer
5390:will produce a stream of
5293:nines' complement '142'.
4362:The decimal repetends of
2650:) and a rational number (
1574:For any integer fraction
564:
435:1.585000... = 1.584999...
15058:Euler's totient function
14772:
13757:
12992:represents the fraction
12438:Extension to other bases
12403: + 1 for some
11703:is prime, the period of
6638:(17)) = LCM(6, 16) = 48,
13820:// the string of digits
13209:is an integer base and
12802: = 1; in the
12262: ≥ 0 we have
11788:, equals the period of
11166:with the first term as
8196:1.000000100000010000001
8166:0.000000100000010000001
7065:,... as defined above.
6597:repeats. An example is
6188:if and only if 10 is a
5246:. See also the article
4627:Fermat's little theorem
2046:{\displaystyle c\neq 0}
1670:
1660:
1653:satisfies the equation
15048:
14942:
14731:Decimal representation
14685:
14551:
14470:
14387:
13663:
13481:
13078:
12982:
12629:
12546:
12353:
12248:
12149:
11821: > 1 and
11750:is strictly less than
11598:
11438:
11309:
11210:is the common factor.
11182:and the common factor
11162:The above series is a
11153:
11131:
11022:
10991:
10969:
10947:
10060:
10029:
10004:
9984:
9959:
9942:
9928:
9903:
9881:
9856:
9834:
9786:= 7, and the smallest
9077:
8673:
8551:
8313:
8112:
8039:
8013:
7989:
7960:
7933:
7904:
7865:
7839:
7800:
7763:
7719:
7693:
7669:
7640:
7613:
7587:
7560:
7407:
7330:
7240:
7147:
6981:
6883:
6642:where LCM denotes the
6408:
6190:primitive root modulo
6123: − 1.
5846:
4778:0434782608695652173913
4483:127, 173... (sequence
4181:2173913043478260869565
3516:0434782608695652173913
2718:
2698:
2644:
2618:
2596:
2519:
2450:
2414:
2363:
2293:
2238:
2209:
2144:
2116:
2075:
2047:
2021:
1993:
1934:
1933:{\displaystyle 10^{n}}
1907:
1887:
1840:
1820:
1800:
1780:
1512:In order to convert a
259:decimal representation
15366:Elementary arithmetic
15049:
14943:
14686:
14552:
14471:
14388:
13664:
13482:
13162:= 0.2 all terminate;
13079:
12983:
12875:of the residue class
12685:lexicographical order
12630:
12547:
12354:
12249:
12150:
11754: − 1.
11725: − 1.
11696: − 1.
11599:
11418:
11310:
11154:
11111:
11023:
10992:
10970:
10948:
10061:
10044:part in the decimal,
10030:
10005:
9990:times, and the digit
9985:
9960:
9940:
9929:
9904:
9882:
9857:
9835:
9078:
8674:
8552:
8314:
8113:
8040:
8014:
7990:
7961:
7934:
7932:{\displaystyle 1000x}
7905:
7866:
7840:
7801:
7764:
7720:
7694:
7670:
7641:
7614:
7588:
7561:
7408:
7331:
7241:
7148:
6982:
6884:
6644:least common multiple
6409:
6346:which states that if
5847:
5574:023255813953488372093
4890:, 96 repeating digits
4868:, 60 repeating digits
4846:, 58 repeating digits
4824:, 46 repeating digits
4802:, 28 repeating digits
4780:, 22 repeating digits
4758:, 18 repeating digits
4736:, 16 repeating digits
4567:Reciprocals of primes
4541:= 1, 2, 3, ..., are:
4478:= 1, 2, 3, ..., are:
4384:= 1, 2, 3, ..., are:
4346:= 1, 2, 3, ..., are:
4290:= 1, 2, 3, ..., are:
4097:023255813953488372093
2719:
2699:
2645:
2619:
2597:
2520:
2430:
2415:
2343:
2273:
2239:
2237:{\displaystyle d_{i}}
2210:
2145:
2117:
2076:
2048:
2022:
1994:
1935:
1908:
1888:
1841:
1821:
1801:
1781:
1545:74 ) 5.00000
15279:Mathematical Gazette
15164:Mathematical Gazette
15056:which again divides
14966:
14859:
14767:Pigeonhole principle
14633:
14486:
14403:
14298:
13531:
13494:
13220:
13213:is an integer, then
12998:
12913:
12860:multiplicative order
12562:
12501:
12419: − 1
12268:
12169:
12047:
11861:multiplicative order
11721:divides evenly into
11614:Transposable integer
11325:
11217:
11047:
11001:
10979:
10959:
10075:
10048:
10014:
9994:
9969:
9949:
9913:
9891:
9866:
9844:
9822:
8693:
8567:
8349:
8148:
8050:
8029:
8012:{\displaystyle =828}
8000:
7988:{\displaystyle 990x}
7976:
7944:
7920:
7876:
7852:
7811:
7790:
7730:
7709:
7680:
7656:
7624:
7600:
7571:
7550:
7354:
7264:
7174:
7081:
6899:
6834:
6369:
6344:Carmichael's theorem
6342:. This follows from
6261:full repetend primes
5802:
5396:pseudo-random digits
4714:, 6 repeating digits
4629:, which states that
2708:
2654:
2628:
2608:
2531:
2426:
2258:
2221:
2156:
2126:
2085:
2057:
2031:
2005:
1946:
1917:
1897:
1850:
1830:
1810:
1790:
1748:
1446:standard uncertainty
518:improve this section
201:improve this article
189:to meet Knowledge's
91:improve this article
15342:"Repeating Decimal"
15259:Dickson, Leonard E.
15143:on 23 December 2023
15131:RoRi (2016-03-01).
15113:on 23 December 2023
13796:"0123..."
13527:
12887:Carmichael function
11624:102564 × 4 = 410256
11620:cyclically permuted
9210:digits 0 after the
7864:{\displaystyle 10x}
7612:{\displaystyle 10x}
7164:are not both zero.
6340:Carmichael function
5398:. Those primes are
2643:{\displaystyle y-c}
2020:{\displaystyle c=0}
1462:countries, such as
1251:11.(1886792452830)
561:
430:1.000... = 0.999...
349:terminating decimal
106:"Repeating decimal"
15339:Weisstein, Eric W.
15230:Albert H. Beiler,
15077:Vuorinen, Aapeli.
15044:
14952:modular arithmetic
14938:
14736:Full reptend prime
14681:
14547:
14466:
14383:
13659:
13477:
13074:
12978:
12806:system there is 0.
12681:order homomorphism
12625:
12542:
12349:
12244:
12145:
11594:
11592:
11305:
11149:
11018:
10987:
10965:
10943:
10941:
10937:
10920:
10903:
10874:
10754:
10737:
10720:
10691:
10571:
10554:
10525:
10405:
10388:
10359:
10239:
10222:
10193:
10056:
10025:
10000:
9980:
9955:
9943:
9934:which is nonzero.
9924:
9899:
9877:
9852:
9830:
9814:In compressed form
9073:
9071:
8669:
8667:
8547:
8545:
8309:
8307:
8108:
8035:
8009:
7985:
7956:
7929:
7900:
7861:
7835:
7796:
7759:
7715:
7692:{\displaystyle =3}
7689:
7668:{\displaystyle 9x}
7665:
7636:
7609:
7583:
7556:
7403:
7326:
7236:
7143:
6977:
6879:
6417:for every integer
6404:
6338:) is known as the
5842:
5366:full reptend prime
5351:full reptend prime
5331:times). They are:
5257:form. For example
5244:{1, 3, 2, 6, 4, 5}
4756:052631578947368421
4525:different cycles (
3960:263157894736842105
3407:052631578947368421
2724:is also rational.
2714:
2694:
2640:
2614:
2592:
2515:
2514:
2410:
2234:
2205:
2140:
2112:
2071:
2043:
2017:
1989:
1930:
1903:
1883:
1836:
1816:
1796:
1776:
1482:irrational numbers
559:
439:division algorithm
261:of a number whose
21:continued fraction
14574:associative array
14378:
14352:
14314:
13656:
13640:
13620:
13600:
13575:
13555:
13535:
13505:
13472:
13469:
13439:
13420:
13367:
13326:
13285:
13244:
13231:
13069:
12965:
12340:
12326:
12301:
12274:
12235:
12208:
12196:
12175:
12139:
12118:
12100:
12079:
12067:
12053:
11692:is always ≤
11588:
11575:
11562:
11534:
11531:
11508:
11487:
11457:
11407:
11387:
11367:
11343:
11303:
11290:
11269:
11266:
11250:
11236:
11147:
11100:
11087:
11074:
11061:
10936:
10919:
10902:
10785:
10753:
10736:
10719:
10602:
10570:
10553:
10436:
10404:
10387:
10270:
10238:
10221:
10104:
10010:will be repeated
10003:{\displaystyle 0}
9965:will be repeated
9958:{\displaystyle 9}
9101:9s. For example,
9067:
9014:
8855:
8753:
8663:
8650:
8623:
8541:
8528:
8480:
8467:
8454:
8425:
8412:
8399:
8303:
8290:
8124:
8123:
8106:
8093:
8064:
8038:{\displaystyle x}
7893:
7890:
7887:
7884:
7828:
7825:
7822:
7819:
7799:{\displaystyle x}
7779:Another example:
7775:
7774:
7757:
7744:
7718:{\displaystyle x}
7559:{\displaystyle x}
7427:The decimal has:
7397:
7320:
7230:
7137:
6877:
6782:is the period of
6755:is the period of
6468:is the period of
5834:
5255:nines' complement
5054:, etc. (sequence
4497:The least primes
4448:The least primes
4217:
4216:
3745:
3744:
3301:
3300:
2717:{\displaystyle x}
2692:
2617:{\displaystyle x}
2587:
2512:
2481:
2394:
2324:
2109:
1983:
1913:. Multiplying by
1906:{\displaystyle b}
1839:{\displaystyle c}
1819:{\displaystyle b}
1799:{\displaystyle a}
1774:
1728:
1727:
1651:= 5.8144144144...
1358:, as well as the
1270:
1269:
554:
553:
546:
255:recurring decimal
251:repeating decimal
247:
246:
239:
229:
228:
221:
191:quality standards
182:This article may
167:
166:
159:
141:
64:
15378:
15352:
15351:
15324:
15314:
15308:
15301:
15295:
15288:
15282:
15275:
15269:
15256:
15250:
15241:
15235:
15228:
15222:
15219:
15213:
15208:Dickson, L. E.,
15206:
15200:
15199:
15158:
15152:
15151:
15149:
15148:
15139:. Archived from
15128:
15122:
15121:
15119:
15118:
15109:. Archived from
15099:
15093:
15092:
15090:
15089:
15074:
15068:
15053:
15051:
15050:
15045:
14999:
14998:
14959:
14947:
14945:
14944:
14939:
14934:
14933:
14918:
14917:
14903:
14871:
14870:
14839:
14833:
14832:
14830:
14829:
14815:
14809:
14808:
14796:
14790:
14783:
14746:Parasitic number
14716:
14714:
14713:
14710:
14707:
14690:
14688:
14687:
14682:
14680:
14679:
14670:
14669:
14660:
14659:
14621:
14619:
14618:
14613:
14610:
14590:
14586:
14582:
14579:. The new digit
14578:
14571:
14567:
14563:
14556:
14554:
14553:
14548:
14542:
14475:
14473:
14472:
14467:
14441:
14392:
14390:
14389:
14384:
14379:
14374:
14366:
14357:
14353:
14348:
14340:
14315:
14310:
14302:
14290:
14283:
14280:the denominator
14276:of the division
14275:
14268:
14261:
14258:
14255:
14252:
14249:
14246:
14243:
14240:
14237:
14234:
14231:
14228:
14225:
14222:
14219:
14216:
14213:
14210:
14207:
14204:
14201:
14198:
14195:
14192:
14189:
14186:
14183:
14180:
14177:
14174:
14171:
14168:
14165:
14162:
14159:
14156:
14153:
14150:
14147:
14144:
14141:
14138:
14135:
14132:
14129:
14126:
14123:
14120:
14117:
14114:
14111:
14108:
14105:
14102:
14099:
14096:
14093:
14090:
14087:
14084:
14081:
14078:
14075:
14072:
14069:
14066:
14063:
14060:
14057:
14054:
14051:
14048:
14045:
14042:
14039:
14036:
14033:
14030:
14027:
14024:
14021:
14018:
14015:
14012:
14009:
14006:
14003:
14000:
13997:
13994:
13991:
13988:
13985:
13982:
13979:
13976:
13973:
13970:
13967:
13964:
13961:
13958:
13955:
13954:
13951:
13948:
13945:
13942:
13938:
13935:
13932:
13929:
13926:
13922:
13921:
13918:
13915:
13912:
13909:
13906:
13903:
13900:
13897:
13894:
13890:
13887:
13884:
13881:
13878:
13875:
13872:
13869:
13866:
13863:
13860:
13857:
13854:
13851:
13848:
13845:
13842:
13839:
13836:
13833:
13830:
13827:
13824:
13821:
13818:
13815:
13812:
13809:
13806:
13803:
13800:
13797:
13794:
13791:
13788:
13785:
13782:
13779:
13776:
13773:
13770:
13767:
13764:
13761:
13754:
13741:
13739:
13737:
13736:
13731:
13728:
13674:
13668:
13666:
13665:
13660:
13658:
13657:
13654:
13652:
13648:
13641:
13639:
13638:
13626:
13621:
13619:
13618:
13606:
13601:
13599:
13598:
13589:
13581:
13576:
13574:
13573:
13561:
13556:
13554:
13553:
13541:
13536:
13534:
13533:
13532:
13517:
13506:
13498:
13486:
13484:
13483:
13478:
13473:
13471:
13470:
13465:
13454:
13442:
13440:
13432:
13421:
13419:
13418:
13409:
13408:
13407:
13379:
13368:
13366:
13365:
13356:
13355:
13354:
13332:
13327:
13325:
13324:
13315:
13314:
13313:
13291:
13286:
13284:
13283:
13274:
13273:
13272:
13250:
13245:
13237:
13232:
13224:
13212:
13208:
13201:
13197:
13195:
13194:
13191:
13188:
13181:
13177:
13175:
13174:
13171:
13168:
13161:
13159:
13158:
13155:
13152:
13145:
13143:
13142:
13139:
13136:
13129:
13127:
13126:
13123:
13120:
13113:
13111:
13110:
13107:
13104:
13094:For example, in
13083:
13081:
13080:
13075:
13070:
13068:
13061:
13060:
13050:
13049:
13048:
13039:
13038:
13026:
13025:
13016:
13015:
13002:
12987:
12985:
12984:
12979:
12977:
12976:
12971:
12967:
12966:
12961:
12960:
12959:
12947:
12946:
12937:
12936:
12926:
12902:
12898:
12884:
12874:
12857:
12850:
12846:
12842:
12838:
12829:
12827:
12826:
12823:
12820:
12813:
12809:
12804:balanced ternary
12801:
12797:
12793:
12777:
12764:
12763:
12732:
12731:
12699:
12674:
12667:
12649:
12647:
12634:
12632:
12631:
12626:
12621:
12620:
12596:
12595:
12583:
12582:
12551:
12549:
12548:
12543:
12514:
12483:
12477:
12429:
12427:
12426:
12423:
12420:
12390:
12388:
12387:
12382:
12379:
12358:
12356:
12355:
12350:
12345:
12341:
12333:
12327:
12324:
12319:
12318:
12306:
12302:
12300:
12299:
12281:
12275:
12272:
12253:
12251:
12250:
12245:
12240:
12236:
12234:
12233:
12215:
12209:
12206:
12201:
12197:
12195:
12194:
12182:
12176:
12173:
12154:
12152:
12151:
12146:
12144:
12140:
12138:
12137:
12125:
12119:
12116:
12105:
12101:
12099:
12098:
12086:
12080:
12077:
12072:
12068:
12060:
12054:
12051:
12029:
12027:
12026:
12021:
12018:
12011:
12009:
12008:
12003:
12000:
11993:
11991:
11990:
11985:
11982:
11964:
11962:
11961:
11956:
11953:
11946:
11944:
11943:
11938:
11935:
11928:
11926:
11925:
11920:
11917:
11910:
11908:
11907:
11895:
11892:
11870:
11842:
11840:
11839:
11834:
11831:
11813: = 2·5
11805:
11803:
11802:
11797:
11794:
11776:
11774:
11773:
11768:
11765:
11749:
11747:
11746:
11741:
11738:
11720:
11718:
11717:
11712:
11709:
11687:
11685:
11684:
11679:
11676:
11657:
11655:
11654:
11651:
11648:
11641:
11639:
11638:
11635:
11632:
11625:
11603:
11601:
11600:
11595:
11593:
11589:
11581:
11576:
11568:
11563:
11561:
11554:
11553:
11540:
11535:
11533:
11532:
11530:
11529:
11517:
11507:
11506:
11494:
11493:
11488:
11486:
11472:
11458:
11456:
11455:
11440:
11437:
11432:
11408:
11406:
11405:
11393:
11388:
11386:
11385:
11373:
11368:
11366:
11365:
11353:
11344:
11336:
11314:
11312:
11311:
11306:
11304:
11296:
11291:
11289:
11275:
11270:
11268:
11267:
11259:
11243:
11242:
11237:
11235:
11221:
11197:
11195:
11194:
11191:
11188:
11181:
11179:
11178:
11175:
11172:
11164:geometric series
11158:
11156:
11155:
11150:
11148:
11146:
11145:
11133:
11130:
11125:
11101:
11093:
11088:
11080:
11075:
11067:
11062:
11054:
11027:
11025:
11024:
11019:
11011:
10996:
10994:
10993:
10988:
10986:
10974:
10972:
10971:
10966:
10952:
10950:
10949:
10944:
10942:
10938:
10929:
10921:
10912:
10904:
10898:
10887:
10879:
10878:
10865:
10849:
10840:
10830:
10817:
10804:
10786:
10778:
10759:
10755:
10746:
10738:
10729:
10721:
10715:
10704:
10696:
10695:
10682:
10666:
10657:
10647:
10634:
10621:
10603:
10595:
10576:
10572:
10563:
10555:
10549:
10538:
10530:
10529:
10516:
10500:
10491:
10481:
10468:
10455:
10437:
10429:
10410:
10406:
10397:
10389:
10383:
10372:
10364:
10363:
10350:
10334:
10325:
10315:
10302:
10289:
10271:
10263:
10244:
10240:
10231:
10223:
10217:
10206:
10198:
10197:
10184:
10168:
10159:
10149:
10136:
10123:
10105:
10097:
10065:
10063:
10062:
10057:
10055:
10034:
10032:
10031:
10026:
10024:
10009:
10007:
10006:
10001:
9989:
9987:
9986:
9981:
9979:
9964:
9962:
9961:
9956:
9933:
9931:
9930:
9925:
9923:
9908:
9906:
9905:
9900:
9898:
9887:its length, and
9886:
9884:
9883:
9878:
9876:
9861:
9859:
9858:
9853:
9851:
9839:
9837:
9836:
9831:
9829:
9810:is therefore 6.
9809:
9807:
9806:
9803:
9800:
9781:
9779:
9778:
9775:
9772:
9754:
9752:
9751:
9746:
9743:
9716:
9714:
9713:
9710:
9707:
9700:
9698:
9697:
9694:
9691:
9681:
9679:
9678:
9675:
9672:
9665:
9663:
9662:
9659:
9656:
9640:
9638:
9637:
9634:
9631:
9624:
9622:
9621:
9618:
9615:
9608:
9606:
9605:
9602:
9599:
9592:
9590:
9589:
9586:
9583:
9576:
9574:
9573:
9570:
9567:
9558:
9556:
9555:
9552:
9549:
9542:
9540:
9539:
9536:
9533:
9526:
9524:
9523:
9520:
9517:
9510:
9508:
9507:
9504:
9501:
9494:
9492:
9491:
9488:
9485:
9478:
9476:
9475:
9472:
9469:
9458:
9456:
9455:
9452:
9449:
9442:
9440:
9439:
9436:
9433:
9426:
9424:
9423:
9420:
9417:
9410:
9408:
9407:
9404:
9401:
9394:
9392:
9391:
9388:
9385:
9376:
9374:
9373:
9370:
9367:
9360:
9358:
9357:
9354:
9351:
9344:
9342:
9341:
9338:
9335:
9328:
9326:
9325:
9322:
9319:
9312:
9310:
9309:
9306:
9303:
9287:
9285:
9284:
9281:
9278:
9271:
9269:
9268:
9265:
9262:
9256:0.00012012... =
9252:
9250:
9249:
9246:
9243:
9233:
9231:
9230:
9227:
9224:
9193:
9191:
9190:
9187:
9184:
9174:
9172:
9171:
9168:
9165:
9158:
9156:
9155:
9152:
9149:
9139:
9137:
9136:
9133:
9130:
9120:
9118:
9117:
9114:
9111:
9082:
9080:
9079:
9074:
9072:
9068:
9066:
9055:
9054:
9053:
9041:
9040:
9031:
9030:
9020:
9015:
9013:
9006:
9005:
8995:
8994:
8993:
8981:
8980:
8971:
8970:
8960:
8944:
8943:
8931:
8930:
8921:
8920:
8886:
8882:
8875:
8874:
8856:
8851:
8850:
8849:
8837:
8836:
8827:
8826:
8816:
8811:
8810:
8798:
8797:
8788:
8787:
8768:
8767:
8754:
8749:
8748:
8747:
8735:
8734:
8725:
8724:
8714:
8678:
8676:
8675:
8670:
8668:
8664:
8656:
8651:
8646:
8629:
8624:
8616:
8602:
8556:
8554:
8553:
8548:
8546:
8542:
8534:
8529:
8527:
8516:
8493:
8485:
8481:
8473:
8468:
8460:
8455:
8453:
8442:
8431:
8426:
8418:
8413:
8405:
8400:
8392:
8384:
8337:
8335:
8334:
8333:10 − 1
8331:
8328:
8318:
8316:
8315:
8310:
8308:
8304:
8296:
8291:
8289:
8282:
8281:
8268:
8230:
8226:
8219:
8218:
8184:
8183:
8141: = 7:
8117:
8115:
8114:
8109:
8107:
8099:
8094:
8092:
8081:
8070:
8065:
8057:
8044:
8042:
8041:
8036:
8018:
8016:
8015:
8010:
7994:
7992:
7991:
7986:
7965:
7963:
7962:
7957:
7938:
7936:
7935:
7930:
7909:
7907:
7906:
7901:
7891:
7888:
7885:
7882:
7870:
7868:
7867:
7862:
7844:
7842:
7841:
7836:
7826:
7823:
7820:
7817:
7805:
7803:
7802:
7797:
7784:
7783:
7768:
7766:
7765:
7760:
7758:
7750:
7745:
7737:
7724:
7722:
7721:
7716:
7698:
7696:
7695:
7690:
7674:
7672:
7671:
7666:
7645:
7643:
7642:
7637:
7618:
7616:
7615:
7610:
7592:
7590:
7589:
7584:
7565:
7563:
7562:
7557:
7544:
7543:
7529:
7527:
7526:
7523:
7520:
7508:
7487:
7483:
7481:
7480:
7477:
7474:
7463:
7461:
7460:
7451:
7448:
7412:
7410:
7409:
7404:
7398:
7396:
7392:
7391:
7382:
7381:
7372:
7371:
7358:
7335:
7333:
7332:
7327:
7321:
7319:
7315:
7314:
7305:
7304:
7295:
7294:
7284:
7283:
7268:
7245:
7243:
7242:
7237:
7231:
7229:
7225:
7224:
7215:
7214:
7205:
7204:
7194:
7193:
7178:
7152:
7150:
7149:
7144:
7138:
7136:
7132:
7131:
7122:
7121:
7112:
7111:
7099:
7098:
7085:
7064:
7062:
7061:
7056:
7053:
7046:
7044:
7043:
7038:
7035:
7028:
7026:
7025:
7020:
7017:
6986:
6984:
6983:
6978:
6967:
6966:
6965:
6964:
6947:
6946:
6945:
6944:
6927:
6926:
6925:
6924:
6888:
6886:
6885:
6880:
6878:
6876:
6872:
6871:
6862:
6861:
6852:
6851:
6838:
6799:
6797:
6796:
6791:
6788:
6772:
6770:
6769:
6764:
6761:
6727:
6725:
6724:
6719:
6716:
6701:
6697:
6695:
6694:
6691:
6688:
6670:
6668:
6667:
6662:
6659:
6612:
6610:
6609:
6606:
6603:
6596:
6594:
6593:
6588:
6585:
6556:
6554:
6553:
6548:
6545:
6531:
6521:
6519:
6518:
6513:
6510:
6503:
6501:
6500:
6495:
6492:
6485:
6483:
6482:
6477:
6474:
6449:
6447:
6446:
6441:
6438:
6413:
6411:
6410:
6405:
6403:
6381:
6380:
6317:
6313:
6311:
6310:
6305:
6302:
6292:
6290:
6289:
6284:
6281:
6242:
6240:
6239:
6234:
6231:
6212:
6187:
6175:totient function
6160:
6158:
6157:
6152:
6149:
6095:
6093:
6092:
6089:
6086:
6077:
6075:
6074:
6071:
6068:
6059:
6057:
6056:
6053:
6050:
6041:
6039:
6038:
6035:
6032:
6023:
6021:
6020:
6017:
6014:
6005:
6003:
6002:
5999:
5996:
5982:
5980:
5979:
5976:
5973:
5964:
5962:
5961:
5958:
5955:
5946:
5944:
5943:
5940:
5937:
5928:
5926:
5925:
5922:
5919:
5910:
5908:
5907:
5904:
5901:
5892:
5890:
5889:
5886:
5883:
5873:
5871:
5870:
5867:
5864:
5851:
5849:
5848:
5843:
5835:
5830:
5823:
5822:
5806:
5794:
5792:
5791:
5786:
5783:
5764:
5762:
5761:
5756:
5753:
5739:
5729:
5725:
5723:
5722:
5719:
5716:
5707:
5703:
5701:
5700:
5697:
5694:
5685:
5681:
5679:
5678:
5675:
5672:
5663:
5659:
5657:
5656:
5653:
5650:
5641:
5637:
5635:
5634:
5631:
5628:
5619:
5615:
5613:
5612:
5609:
5606:
5597:
5593:
5591:
5590:
5587:
5584:
5575:
5571:
5569:
5568:
5565:
5562:
5553:
5549:
5547:
5546:
5543:
5540:
5531:
5527:
5525:
5524:
5521:
5518:
5509:
5505:
5503:
5502:
5499:
5496:
5487:
5483:
5481:
5480:
5477:
5474:
5465:
5461:
5459:
5458:
5455:
5452:
5443:
5439:
5437:
5436:
5433:
5430:
5407:
5389:
5387:
5386:
5381:
5378:
5340:
5330:
5328:
5327:
5324:
5321:
5288:
5286:
5285:
5282:
5279:
5272:
5270:
5269:
5266:
5263:
5245:
5241:
5239:
5238:
5235:
5232:
5222:
5218:
5214:
5212:
5211:
5208:
5205:
5197:
5193:
5189:
5187:
5186:
5183:
5180:
5172:
5168:
5164:
5162:
5161:
5158:
5155:
5147:
5143:
5139:
5137:
5136:
5133:
5130:
5122:
5118:
5114:
5112:
5111:
5108:
5105:
5097:
5093:
5089:
5087:
5086:
5083:
5080:
5059:
5053:
5051:
5050:
5047:
5044:
5037:
5035:
5034:
5031:
5028:
5021:
5019:
5018:
5015:
5012:
5005:
5003:
5002:
4999:
4996:
4989:
4987:
4986:
4983:
4980:
4973:
4971:
4970:
4967:
4964:
4957:
4955:
4954:
4951:
4948:
4941:
4939:
4938:
4935:
4932:
4925:
4923:
4922:
4919:
4916:
4909:
4907:
4906:
4903:
4900:
4889:
4885:
4883:
4882:
4879:
4876:
4867:
4863:
4861:
4860:
4857:
4854:
4845:
4841:
4839:
4838:
4835:
4832:
4823:
4819:
4817:
4816:
4813:
4810:
4801:
4797:
4795:
4794:
4791:
4788:
4779:
4775:
4773:
4772:
4769:
4766:
4757:
4753:
4751:
4750:
4747:
4744:
4735:
4734:0588235294117647
4731:
4729:
4728:
4725:
4722:
4713:
4709:
4707:
4706:
4703:
4700:
4664:
4662:
4661:
4656:
4653:
4636:
4601:is equal to the
4600:
4598:
4597:
4592:
4589:
4551:
4536:
4520:
4518:
4517:
4512:
4509:
4488:
4469:
4467:
4466:
4461:
4458:
4439:
4429:th prime), are:
4425:= 2, 3, 5, ... (
4420:
4418:
4417:
4412:
4409:
4394:
4379:
4377:
4376:
4371:
4368:
4341:
4339:
4338:
4333:
4330:
4300:
4285:
4283:
4282:
4277:
4274:
4246:
4244:
4243:
4238:
4235:
4210:
4204:
4202:
4201:
4198:
4195:
4182:
4176:
4174:
4173:
4170:
4167:
4154:
4148:
4146:
4145:
4142:
4139:
4126:
4120:
4118:
4117:
4114:
4111:
4098:
4092:
4090:
4089:
4086:
4083:
4070:
4064:
4062:
4061:
4058:
4055:
4042:
4036:
4034:
4033:
4030:
4027:
4011:
4009:
4008:
4005:
4002:
3989:
3983:
3981:
3980:
3977:
3974:
3961:
3955:
3953:
3952:
3949:
3946:
3933:
3927:
3925:
3924:
3921:
3918:
3905:
3899:
3897:
3896:
3893:
3890:
3877:
3871:
3869:
3868:
3865:
3862:
3849:
3848:2941176470588235
3843:
3841:
3840:
3837:
3834:
3821:
3815:
3813:
3812:
3809:
3806:
3790:
3788:
3787:
3784:
3781:
3771:
3749:
3748:
3738:
3732:
3730:
3729:
3726:
3723:
3710:
3704:
3702:
3701:
3698:
3695:
3682:
3676:
3674:
3673:
3670:
3667:
3654:
3648:
3646:
3645:
3642:
3639:
3626:
3620:
3618:
3617:
3614:
3611:
3598:
3592:
3590:
3589:
3586:
3583:
3567:
3565:
3564:
3561:
3558:
3545:
3539:
3537:
3536:
3533:
3530:
3517:
3511:
3509:
3508:
3505:
3502:
3489:
3483:
3481:
3480:
3477:
3474:
3461:
3455:
3453:
3452:
3449:
3446:
3430:
3428:
3427:
3424:
3421:
3408:
3402:
3400:
3399:
3396:
3393:
3380:
3374:
3372:
3371:
3368:
3365:
3352:
3351:0588235294117647
3346:
3344:
3343:
3340:
3337:
3327:
3305:
3304:
3285:
3283:
3282:
3279:
3276:
3263:
3254:
3248:
3246:
3245:
3242:
3239:
3226:
3217:
3211:
3209:
3208:
3205:
3202:
3189:
3180:
3174:
3172:
3171:
3168:
3165:
3152:
3143:
3137:
3135:
3134:
3131:
3128:
3115:
3106:
3100:
3098:
3097:
3094:
3091:
3078:
3066:
3064:
3063:
3060:
3057:
3044:
3035:
3029:
3027:
3026:
3023:
3020:
2998:
2996:
2995:
2992:
2989:
2976:
2967:
2961:
2959:
2958:
2955:
2952:
2939:
2930:
2924:
2922:
2921:
2918:
2915:
2902:
2890:
2888:
2887:
2884:
2881:
2859:
2857:
2856:
2853:
2850:
2837:
2828:
2822:
2820:
2819:
2816:
2813:
2791:
2789:
2788:
2785:
2782:
2772:
2757:
2735:
2734:
2723:
2721:
2720:
2715:
2703:
2701:
2700:
2695:
2693:
2691:
2684:
2683:
2673:
2669:
2668:
2658:
2649:
2647:
2646:
2641:
2623:
2621:
2620:
2615:
2601:
2599:
2598:
2593:
2588:
2586:
2579:
2578:
2568:
2564:
2563:
2553:
2524:
2522:
2521:
2516:
2513:
2511:
2510:
2509:
2487:
2482:
2480:
2479:
2474:
2470:
2469:
2452:
2449:
2444:
2419:
2417:
2416:
2411:
2400:
2396:
2395:
2393:
2392:
2387:
2383:
2382:
2365:
2362:
2357:
2325:
2323:
2322:
2317:
2313:
2312:
2295:
2292:
2287:
2243:
2241:
2240:
2235:
2233:
2232:
2214:
2212:
2211:
2206:
2204:
2203:
2184:
2183:
2174:
2173:
2149:
2147:
2146:
2141:
2139:
2121:
2119:
2118:
2113:
2111:
2110:
2102:
2080:
2078:
2077:
2072:
2070:
2052:
2050:
2049:
2044:
2026:
2024:
2023:
2018:
1998:
1996:
1995:
1990:
1985:
1984:
1976:
1958:
1957:
1939:
1937:
1936:
1931:
1929:
1928:
1912:
1910:
1909:
1904:
1892:
1890:
1889:
1884:
1879:
1872:
1871:
1845:
1843:
1842:
1837:
1825:
1823:
1822:
1817:
1805:
1803:
1802:
1797:
1785:
1783:
1782:
1777:
1775:
1767:
1724:
1722:
1721:
1718:
1715:
1708:
1706:
1705:
1702:
1699:
1658:
1657:
1652:
1593:
1591:
1590:
1585:
1582:
1570:
1569:
1566:
1543:
1535:
1533:
1532:
1529:
1526:
1502:
1497:
1488:
1265:
1264:
1261:11.1886792452830
1257:
1248:
1247:
1246:
1243:
1237:
1236:
1235:
1232:
1224:
1218:
1216:
1215:
1212:
1209:
1200:
1198:
1197:
1194:
1191:
1180:
1179:
1172:
1163:
1162:
1161:
1158:
1152:
1151:
1150:
1147:
1139:
1133:
1131:
1130:
1127:
1124:
1115:
1113:
1112:
1109:
1106:
1095:
1094:
1087:
1078:
1077:
1076:
1073:
1067:
1066:
1065:
1062:
1054:
1048:
1046:
1045:
1042:
1039:
1030:
1028:
1027:
1024:
1021:
1010:
1009:
1002:
993:
992:
991:
988:
982:
981:
980:
977:
969:
963:
961:
960:
957:
954:
945:
943:
942:
939:
936:
925:
924:
917:
908:
907:
906:
903:
895:
889:
887:
886:
883:
880:
871:
869:
868:
865:
862:
849:
840:
839:
838:
835:
830:
829:
828:
825:
817:
811:
809:
808:
805:
802:
793:
791:
790:
787:
784:
771:
762:
761:
760:
757:
749:
743:
741:
740:
737:
734:
725:
723:
722:
719:
716:
703:
694:
693:
692:
689:
681:
675:
673:
672:
669:
666:
657:
655:
654:
651:
648:
635:
626:
625:
624:
621:
613:
605:
603:
602:
599:
596:
562:
558:
549:
542:
538:
535:
529:
498:
490:
475:
469:
468:
467:
436:
431:
418:
417:
415:
414:
411:
408:
399:
397:
396:
393:
390:
377:
376:
374:
373:
370:
367:
353:decimal fraction
335:
333:
332:
329:
326:
315:
313:
312:
309:
306:
295:
293:
292:
289:
286:
242:
235:
224:
217:
213:
210:
204:
177:
176:
169:
162:
155:
151:
148:
142:
140:
99:
75:
67:
56:
34:
33:
26:
15386:
15385:
15381:
15380:
15379:
15377:
15376:
15375:
15371:Numeral systems
15356:
15355:
15333:
15328:
15327:
15315:
15311:
15302:
15298:
15289:
15285:
15276:
15272:
15257:
15253:
15242:
15238:
15229:
15225:
15220:
15216:
15207:
15203:
15177:10.2307/3621484
15159:
15155:
15146:
15144:
15137:Stumbling Robot
15129:
15125:
15116:
15114:
15101:
15100:
15096:
15087:
15085:
15083:Aapeli Vuorinen
15075:
15071:
14994:
14990:
14967:
14964:
14963:
14954:
14929:
14925:
14913:
14909:
14899:
14866:
14862:
14860:
14857:
14856:
14840:
14836:
14827:
14825:
14817:
14816:
14812:
14797:
14793:
14784:
14780:
14775:
14727:
14711:
14708:
14702:
14701:
14699:
14675:
14671:
14665:
14661:
14655:
14651:
14634:
14631:
14630:
14626:) is given by:
14614:
14611:
14608:
14607:
14605:
14601:
14588:
14584:
14580:
14576:
14569:
14565:
14561:
14535:
14487:
14484:
14483:
14434:
14404:
14401:
14400:
14367:
14365:
14341:
14339:
14335:
14303:
14301:
14299:
14296:
14295:
14288:
14281:
14273:
14266:
14263:
14262:
14259:
14256:
14253:
14250:
14247:
14244:
14241:
14238:
14235:
14232:
14229:
14226:
14223:
14220:
14217:
14214:
14211:
14208:
14205:
14202:
14199:
14196:
14193:
14190:
14187:
14184:
14181:
14178:
14175:
14172:
14169:
14166:
14163:
14160:
14157:
14154:
14151:
14148:
14145:
14142:
14139:
14136:
14133:
14130:
14127:
14124:
14121:
14118:
14115:
14112:
14109:
14106:
14103:
14100:
14097:
14094:
14091:
14088:
14085:
14082:
14079:
14076:
14073:
14070:
14067:
14064:
14061:
14058:
14055:
14052:
14049:
14046:
14043:
14040:
14037:
14034:
14031:
14028:
14025:
14022:
14019:
14016:
14013:
14010:
14007:
14004:
14001:
13998:
13995:
13992:
13989:
13986:
13983:
13980:
13977:
13974:
13971:
13968:
13965:
13962:
13959:
13956:
13953:// 0 ≤ p < q
13952:
13949:
13946:
13943:
13940:
13936:
13933:
13930:
13927:
13924:
13923:
13919:
13916:
13913:
13910:
13907:
13904:
13901:
13898:
13895:
13892:
13891:
13888:
13885:
13882:
13879:
13876:
13873:
13870:
13867:
13864:
13861:
13858:
13855:
13852:
13849:
13846:
13843:
13840:
13837:
13834:
13831:
13828:
13825:
13822:
13819:
13816:
13813:
13810:
13807:
13804:
13801:
13798:
13795:
13792:
13789:
13786:
13783:
13780:
13777:
13774:
13771:
13768:
13765:
13762:
13759:
13753:
13743:
13732:
13729:
13724:
13723:
13721:
13719:
13718:For a rational
13716:
13709:
13705:
13701:
13697:
13693:
13689:
13685:
13681:
13677:
13672:
13653:
13634:
13630:
13625:
13614:
13610:
13605:
13594:
13590:
13582:
13580:
13569:
13565:
13560:
13549:
13545:
13540:
13526:
13525:
13521:
13516:
13515:
13511:
13510:
13497:
13495:
13492:
13491:
13455:
13453:
13446:
13441:
13431:
13414:
13410:
13397:
13393:
13380:
13378:
13361:
13357:
13350:
13346:
13333:
13331:
13320:
13316:
13309:
13305:
13292:
13290:
13279:
13275:
13268:
13264:
13251:
13249:
13236:
13223:
13221:
13218:
13217:
13210:
13206:
13199:
13192:
13189:
13186:
13185:
13183:
13179:
13172:
13169:
13166:
13165:
13163:
13156:
13153:
13150:
13149:
13147:
13140:
13137:
13134:
13133:
13131:
13124:
13121:
13118:
13117:
13115:
13108:
13105:
13102:
13101:
13099:
13056:
13052:
13051:
13044:
13040:
13034:
13030:
13021:
13017:
13011:
13007:
13003:
13001:
12999:
12996:
12995:
12972:
12955:
12951:
12942:
12938:
12932:
12928:
12927:
12925:
12921:
12917:
12916:
12914:
12911:
12910:
12900:
12889:
12876:
12868:
12862:
12852:
12848:
12844:
12840:
12836:
12824:
12821:
12818:
12817:
12815:
12811:
12810: = 1.
12807:
12799:
12798: = 1.
12795:
12791:
12784:
12779:
12771:
12766:
12762:
12756:
12753:
12747:
12741:
12734:
12729:
12724:
12723:
12714:
12708:
12701:
12695:
12677:non-terminating
12669:
12661:
12656:
12651:
12645:
12644: := |
12640:
12616:
12612:
12591:
12587:
12578:
12574:
12563:
12560:
12559:
12510:
12502:
12499:
12498:
12479:
12473:
12440:
12424:
12421:
12415:
12414:
12412:
12383:
12380:
12377:
12376:
12374:
12332:
12328:
12323:
12314:
12310:
12289:
12285:
12280:
12276:
12271:
12269:
12266:
12265:
12223:
12219:
12214:
12210:
12205:
12190:
12186:
12181:
12177:
12172:
12170:
12167:
12166:
12133:
12129:
12124:
12120:
12115:
12094:
12090:
12085:
12081:
12076:
12059:
12055:
12050:
12048:
12045:
12044:
12022:
12019:
12016:
12015:
12013:
12004:
12001:
11998:
11997:
11995:
11986:
11983:
11980:
11979:
11977:
11957:
11954:
11951:
11950:
11948:
11939:
11936:
11933:
11932:
11930:
11921:
11918:
11915:
11914:
11912:
11896:
11893:
11890:
11889:
11887:
11864:
11835:
11832:
11829:
11828:
11826:
11798:
11795:
11792:
11791:
11789:
11769:
11766:
11761:
11760:
11758:
11742:
11739:
11736:
11735:
11733:
11713:
11710:
11707:
11706:
11704:
11680:
11677:
11674:
11673:
11671:
11664:
11652:
11649:
11646:
11645:
11643:
11636:
11633:
11630:
11629:
11627:
11623:
11616:
11610:
11591:
11590:
11580:
11567:
11549:
11545:
11544:
11539:
11525:
11521:
11516:
11509:
11502:
11498:
11492:
11476:
11471:
11468:
11460:
11459:
11448:
11444:
11439:
11433:
11422:
11401:
11397:
11392:
11381:
11377:
11372:
11361:
11357:
11352:
11345:
11335:
11328:
11326:
11323:
11322:
11295:
11279:
11274:
11258:
11251:
11241:
11225:
11220:
11218:
11215:
11214:
11192:
11189:
11186:
11185:
11183:
11176:
11173:
11170:
11169:
11167:
11141:
11137:
11132:
11126:
11115:
11092:
11079:
11066:
11053:
11048:
11045:
11044:
11038:infinite series
11034:
11007:
11002:
10999:
10998:
10982:
10980:
10977:
10976:
10960:
10957:
10956:
10940:
10939:
10927:
10922:
10910:
10905:
10888:
10885:
10880:
10873:
10872:
10861:
10856:
10845:
10838:
10837:
10826:
10824:
10813:
10811:
10800:
10793:
10792:
10787:
10777:
10769:
10760:
10757:
10756:
10744:
10739:
10727:
10722:
10705:
10702:
10697:
10690:
10689:
10678:
10673:
10662:
10655:
10654:
10643:
10641:
10630:
10628:
10617:
10610:
10609:
10604:
10594:
10586:
10577:
10574:
10573:
10561:
10556:
10539:
10536:
10531:
10524:
10523:
10512:
10507:
10496:
10489:
10488:
10477:
10475:
10464:
10462:
10451:
10444:
10443:
10438:
10428:
10420:
10411:
10408:
10407:
10395:
10390:
10373:
10370:
10365:
10358:
10357:
10346:
10341:
10330:
10323:
10322:
10311:
10309:
10298:
10296:
10285:
10278:
10277:
10272:
10262:
10254:
10245:
10242:
10241:
10229:
10224:
10207:
10204:
10199:
10192:
10191:
10180:
10175:
10164:
10157:
10156:
10145:
10143:
10132:
10130:
10119:
10112:
10111:
10106:
10096:
10088:
10078:
10076:
10073:
10072:
10051:
10049:
10046:
10045:
10020:
10015:
10012:
10011:
9995:
9992:
9991:
9975:
9970:
9967:
9966:
9950:
9947:
9946:
9919:
9914:
9911:
9910:
9894:
9892:
9889:
9888:
9872:
9867:
9864:
9863:
9847:
9845:
9842:
9841:
9825:
9823:
9820:
9819:
9816:
9804:
9801:
9798:
9797:
9795:
9776:
9773:
9770:
9769:
9767:
9747:
9744:
9739:
9738:
9736:
9711:
9708:
9705:
9704:
9702:
9695:
9692:
9689:
9688:
9686:
9685:0.3789789... =
9676:
9673:
9670:
9669:
9667:
9660:
9657:
9654:
9653:
9651:
9635:
9632:
9629:
9628:
9626:
9619:
9616:
9613:
9612:
9610:
9603:
9600:
9597:
9596:
9594:
9587:
9584:
9581:
9580:
9578:
9571:
9568:
9565:
9564:
9562:
9553:
9550:
9547:
9546:
9544:
9537:
9534:
9531:
9530:
9528:
9521:
9518:
9515:
9514:
9512:
9505:
9502:
9499:
9498:
9496:
9489:
9486:
9483:
9482:
9480:
9473:
9470:
9467:
9466:
9464:
9453:
9450:
9447:
9446:
9444:
9437:
9434:
9431:
9430:
9428:
9421:
9418:
9415:
9414:
9412:
9405:
9402:
9399:
9398:
9396:
9389:
9386:
9383:
9382:
9380:
9371:
9368:
9365:
9364:
9362:
9355:
9352:
9349:
9348:
9346:
9339:
9336:
9333:
9332:
9330:
9323:
9320:
9317:
9316:
9314:
9307:
9304:
9301:
9300:
9298:
9282:
9279:
9276:
9275:
9273:
9266:
9263:
9260:
9259:
9257:
9247:
9244:
9241:
9240:
9238:
9228:
9225:
9222:
9221:
9219:
9188:
9185:
9182:
9181:
9179:
9169:
9166:
9163:
9162:
9160:
9153:
9150:
9147:
9146:
9144:
9134:
9131:
9128:
9127:
9125:
9115:
9112:
9109:
9108:
9106:
9070:
9069:
9056:
9049:
9045:
9036:
9032:
9026:
9022:
9021:
9019:
9001:
8997:
8996:
8989:
8985:
8976:
8972:
8966:
8962:
8961:
8959:
8952:
8946:
8945:
8939:
8935:
8926:
8922:
8916:
8912:
8905:
8870:
8866:
8865:
8861:
8858:
8857:
8845:
8841:
8832:
8828:
8822:
8818:
8817:
8815:
8806:
8802:
8793:
8789:
8783:
8779:
8772:
8763:
8759:
8756:
8755:
8743:
8739:
8730:
8726:
8720:
8716:
8715:
8713:
8703:
8696:
8694:
8691:
8690:
8666:
8665:
8655:
8630:
8628:
8615:
8600:
8599:
8580:
8570:
8568:
8565:
8564:
8544:
8543:
8533:
8517:
8494:
8492:
8483:
8482:
8472:
8459:
8443:
8432:
8430:
8417:
8404:
8391:
8382:
8381:
8362:
8352:
8350:
8347:
8346:
8332:
8329:
8326:
8325:
8323:
8306:
8305:
8295:
8277:
8273:
8272:
8267:
8260:
8254:
8253:
8243:
8214:
8210:
8209:
8205:
8202:
8201:
8188:
8179:
8175:
8172:
8171:
8158:
8151:
8149:
8146:
8145:
8131:
8098:
8082:
8071:
8069:
8056:
8051:
8048:
8047:
8030:
8027:
8026:
8001:
7998:
7997:
7977:
7974:
7973:
7945:
7942:
7941:
7921:
7918:
7917:
7877:
7874:
7873:
7853:
7850:
7849:
7812:
7809:
7808:
7791:
7788:
7787:
7749:
7736:
7731:
7728:
7727:
7710:
7707:
7706:
7681:
7678:
7677:
7657:
7654:
7653:
7625:
7622:
7621:
7601:
7598:
7597:
7572:
7569:
7568:
7551:
7548:
7547:
7537:
7524:
7521:
7518:
7517:
7515:
7500:
7485:
7478:
7475:
7472:
7471:
7469:
7452:
7449:
7446:
7445:
7443:
7387:
7383:
7377:
7373:
7367:
7363:
7362:
7357:
7355:
7352:
7351:
7310:
7306:
7300:
7296:
7290:
7286:
7285:
7273:
7269:
7267:
7265:
7262:
7261:
7220:
7216:
7210:
7206:
7200:
7196:
7195:
7183:
7179:
7177:
7175:
7172:
7171:
7127:
7123:
7117:
7113:
7107:
7103:
7094:
7090:
7089:
7084:
7082:
7079:
7078:
7071:
7057:
7054:
7051:
7050:
7048:
7039:
7036:
7033:
7032:
7030:
7021:
7018:
7015:
7014:
7012:
7009:
7002:
6995:
6960:
6956:
6955:
6951:
6940:
6936:
6935:
6931:
6920:
6916:
6915:
6911:
6900:
6897:
6896:
6867:
6863:
6857:
6853:
6847:
6843:
6842:
6837:
6835:
6832:
6831:
6792:
6789:
6786:
6785:
6783:
6781:
6765:
6762:
6759:
6758:
6756:
6754:
6745:
6736:
6720:
6717:
6714:
6713:
6711:
6699:
6692:
6689:
6686:
6685:
6683:
6671:is a factor of
6663:
6660:
6657:
6656:
6654:
6607:
6604:
6601:
6600:
6598:
6589:
6586:
6583:
6582:
6580:
6568:
6549:
6546:
6543:
6542:
6540:
6527:
6514:
6511:
6508:
6507:
6505:
6496:
6493:
6490:
6489:
6487:
6478:
6475:
6472:
6471:
6469:
6467:
6458:
6442:
6439:
6436:
6435:
6433:
6388:
6376:
6372:
6370:
6367:
6366:
6315:
6306:
6303:
6300:
6299:
6297:
6285:
6282:
6279:
6278:
6276:
6269:
6235:
6232:
6227:
6226:
6224:
6199:
6178:
6153:
6150:
6147:
6146:
6144:
6129:
6090:
6087:
6084:
6083:
6081:
6072:
6069:
6066:
6065:
6063:
6054:
6051:
6048:
6047:
6045:
6036:
6033:
6030:
6029:
6027:
6018:
6015:
6012:
6011:
6009:
6000:
5997:
5994:
5993:
5991:
5977:
5974:
5971:
5970:
5968:
5959:
5956:
5953:
5952:
5950:
5941:
5938:
5935:
5934:
5932:
5923:
5920:
5917:
5916:
5914:
5905:
5902:
5899:
5898:
5896:
5887:
5884:
5881:
5880:
5878:
5868:
5865:
5862:
5861:
5859:
5812:
5808:
5807:
5805:
5803:
5800:
5799:
5787:
5784:
5781:
5780:
5778:
5757:
5754:
5751:
5750:
5748:
5735:
5727:
5720:
5717:
5714:
5713:
5711:
5705:
5698:
5695:
5692:
5691:
5689:
5683:
5676:
5673:
5670:
5669:
5667:
5661:
5654:
5651:
5648:
5647:
5645:
5639:
5632:
5629:
5626:
5625:
5623:
5617:
5610:
5607:
5604:
5603:
5601:
5595:
5588:
5585:
5582:
5581:
5579:
5573:
5566:
5563:
5560:
5559:
5557:
5551:
5544:
5541:
5538:
5537:
5535:
5529:
5522:
5519:
5516:
5515:
5513:
5508:032258064516129
5507:
5500:
5497:
5494:
5493:
5491:
5485:
5478:
5475:
5472:
5471:
5469:
5463:
5456:
5453:
5450:
5449:
5447:
5441:
5434:
5431:
5428:
5427:
5425:
5419:
5403:
5394: − 1
5382:
5379:
5376:
5375:
5373:
5336:
5325:
5322:
5316:
5315:
5313:
5283:
5280:
5277:
5276:
5274:
5267:
5264:
5261:
5260:
5258:
5243:
5236:
5233:
5230:
5229:
5227:
5220:
5216:
5209:
5206:
5203:
5202:
5200:
5195:
5191:
5184:
5181:
5178:
5177:
5175:
5170:
5166:
5159:
5156:
5153:
5152:
5150:
5145:
5141:
5134:
5131:
5128:
5127:
5125:
5120:
5116:
5109:
5106:
5103:
5102:
5100:
5095:
5091:
5084:
5081:
5078:
5077:
5075:
5055:
5048:
5045:
5042:
5041:
5039:
5032:
5029:
5026:
5025:
5023:
5016:
5013:
5010:
5009:
5007:
5000:
4997:
4994:
4993:
4991:
4984:
4981:
4978:
4977:
4975:
4968:
4965:
4962:
4961:
4959:
4952:
4949:
4946:
4945:
4943:
4936:
4933:
4930:
4929:
4927:
4920:
4917:
4914:
4913:
4911:
4904:
4901:
4898:
4897:
4895:
4887:
4880:
4877:
4874:
4873:
4871:
4865:
4858:
4855:
4852:
4851:
4849:
4843:
4836:
4833:
4830:
4829:
4827:
4821:
4814:
4811:
4808:
4807:
4805:
4799:
4792:
4789:
4786:
4785:
4783:
4777:
4770:
4767:
4764:
4763:
4761:
4755:
4748:
4745:
4742:
4741:
4739:
4733:
4726:
4723:
4720:
4719:
4717:
4711:
4704:
4701:
4698:
4697:
4695:
4689:
4683:
4657:
4654:
4651:
4650:
4648:
4630:
4593:
4590:
4587:
4586:
4584:
4573:in lowest terms
4569:
4563:
4547:
4526:
4513:
4510:
4505:
4504:
4502:
4484:
4462:
4459:
4456:
4455:
4453:
4435:
4413:
4410:
4407:
4406:
4404:
4390:
4372:
4369:
4366:
4365:
4363:
4334:
4331:
4328:
4327:
4325:
4315:
4296:
4278:
4275:
4272:
4271:
4269:
4263:
4253:
4239:
4236:
4233:
4232:
4230:
4220:
4208:
4199:
4196:
4193:
4192:
4190:
4180:
4171:
4168:
4165:
4164:
4162:
4152:
4143:
4140:
4137:
4136:
4134:
4124:
4115:
4112:
4109:
4108:
4106:
4096:
4087:
4084:
4081:
4080:
4078:
4068:
4059:
4056:
4053:
4052:
4050:
4040:
4031:
4028:
4025:
4024:
4022:
4006:
4003:
4000:
3999:
3997:
3987:
3978:
3975:
3972:
3971:
3969:
3959:
3950:
3947:
3944:
3943:
3941:
3931:
3922:
3919:
3916:
3915:
3913:
3903:
3894:
3891:
3888:
3887:
3885:
3875:
3866:
3863:
3860:
3859:
3857:
3847:
3838:
3835:
3832:
3831:
3829:
3819:
3810:
3807:
3804:
3803:
3801:
3785:
3782:
3779:
3778:
3776:
3770:
3764:
3760:
3756:
3737:032258064516129
3736:
3727:
3724:
3721:
3720:
3718:
3708:
3699:
3696:
3693:
3692:
3690:
3680:
3671:
3668:
3665:
3664:
3662:
3652:
3643:
3640:
3637:
3636:
3634:
3624:
3615:
3612:
3609:
3608:
3606:
3596:
3587:
3584:
3581:
3580:
3578:
3562:
3559:
3556:
3555:
3553:
3543:
3534:
3531:
3528:
3527:
3525:
3515:
3506:
3503:
3500:
3499:
3497:
3487:
3478:
3475:
3472:
3471:
3469:
3459:
3450:
3447:
3444:
3443:
3441:
3425:
3422:
3419:
3418:
3416:
3406:
3397:
3394:
3391:
3390:
3388:
3378:
3369:
3366:
3363:
3362:
3360:
3350:
3341:
3338:
3335:
3334:
3332:
3326:
3320:
3316:
3312:
3280:
3277:
3274:
3273:
3271:
3261:
3252:
3243:
3240:
3237:
3236:
3234:
3224:
3215:
3206:
3203:
3200:
3199:
3197:
3187:
3178:
3169:
3166:
3163:
3162:
3160:
3150:
3141:
3132:
3129:
3126:
3125:
3123:
3113:
3104:
3095:
3092:
3089:
3088:
3086:
3076:
3061:
3058:
3055:
3054:
3052:
3042:
3033:
3024:
3021:
3018:
3017:
3015:
2993:
2990:
2987:
2986:
2984:
2974:
2965:
2956:
2953:
2950:
2949:
2947:
2937:
2928:
2919:
2916:
2913:
2912:
2910:
2900:
2885:
2882:
2879:
2878:
2876:
2854:
2851:
2848:
2847:
2845:
2835:
2826:
2817:
2814:
2811:
2810:
2808:
2786:
2783:
2780:
2779:
2777:
2771:
2765:
2761:
2756:
2750:
2746:
2742:
2730:
2728:Table of values
2709:
2706:
2705:
2679:
2675:
2674:
2664:
2660:
2659:
2657:
2655:
2652:
2651:
2629:
2626:
2625:
2609:
2606:
2605:
2574:
2570:
2569:
2559:
2555:
2554:
2552:
2532:
2529:
2528:
2502:
2498:
2491:
2486:
2475:
2465:
2461:
2457:
2456:
2451:
2445:
2434:
2427:
2424:
2423:
2388:
2378:
2374:
2370:
2369:
2364:
2358:
2347:
2339:
2335:
2318:
2308:
2304:
2300:
2299:
2294:
2288:
2277:
2259:
2256:
2255:
2228:
2224:
2222:
2219:
2218:
2199:
2195:
2179:
2175:
2169:
2165:
2157:
2154:
2153:
2135:
2127:
2124:
2123:
2101:
2100:
2086:
2083:
2082:
2066:
2058:
2055:
2054:
2032:
2029:
2028:
2006:
2003:
2002:
1975:
1974:
1953:
1949:
1947:
1944:
1943:
1924:
1920:
1918:
1915:
1914:
1898:
1895:
1894:
1867:
1863:
1862:
1851:
1848:
1847:
1831:
1828:
1827:
1811:
1808:
1807:
1791:
1788:
1787:
1766:
1749:
1746:
1745:
1742:
1719:
1716:
1713:
1712:
1710:
1703:
1700:
1697:
1696:
1694:
1647:
1644:linear equation
1640:
1624:in lowest terms
1612:
1586:
1583:
1578:
1577:
1575:
1567:
1564:
1562:
1558:
1553:420
1541:
1530:
1527:
1524:
1523:
1521:
1514:rational number
1510:
1500:
1495:
1486:
1262:
1260:
1255:
1244:
1241:
1240:
1239:
1233:
1230:
1229:
1228:
1222:
1213:
1210:
1208:111886792452819
1207:
1206:
1204:
1195:
1192:
1189:
1188:
1186:
1177:
1175:
1170:
1159:
1156:
1155:
1154:
1148:
1145:
1144:
1143:
1137:
1128:
1125:
1122:
1121:
1119:
1110:
1107:
1104:
1103:
1101:
1092:
1090:
1085:
1074:
1071:
1070:
1069:
1063:
1060:
1059:
1058:
1052:
1043:
1040:
1037:
1036:
1034:
1025:
1022:
1019:
1018:
1016:
1007:
1005:
1000:
989:
986:
985:
984:
978:
975:
974:
973:
967:
958:
955:
952:
951:
949:
940:
937:
934:
933:
931:
922:
920:
915:
904:
901:
900:
899:
893:
884:
881:
878:
877:
875:
866:
863:
860:
859:
857:
847:
836:
833:
832:
831:
826:
823:
822:
821:
815:
806:
803:
800:
799:
797:
788:
785:
782:
781:
779:
769:
758:
755:
754:
753:
747:
738:
735:
732:
731:
729:
720:
717:
714:
713:
711:
701:
690:
687:
686:
685:
679:
670:
667:
664:
663:
661:
652:
649:
646:
645:
643:
633:
622:
619:
618:
617:
611:
600:
597:
594:
593:
591:
550:
539:
533:
530:
515:
499:
488:
483:
473:
465:
463:
462:
434:
429:
412:
409:
406:
405:
403:
401:
394:
391:
386:
385:
383:
371:
368:
365:
364:
362:
360:
330:
327:
324:
323:
321:
310:
307:
304:
303:
301:
290:
287:
284:
283:
281:
265:are eventually
243:
232:
231:
230:
225:
214:
208:
205:
198:
178:
174:
163:
152:
146:
143:
100:
98:
88:
76:
35:
31:
24:
17:
12:
11:
5:
15384:
15374:
15373:
15368:
15354:
15353:
15332:
15331:External links
15329:
15326:
15325:
15309:
15296:
15283:
15270:
15251:
15236:
15223:
15214:
15201:
15153:
15123:
15094:
15069:
15055:
15054:
15043:
15040:
15037:
15034:
15031:
15028:
15025:
15022:
15019:
15015:
15011:
15008:
15005:
15002:
14997:
14993:
14989:
14986:
14983:
14980:
14977:
14974:
14971:
14949:
14948:
14937:
14932:
14928:
14924:
14921:
14916:
14912:
14907:
14902:
14898:
14895:
14892:
14889:
14886:
14883:
14880:
14877:
14874:
14869:
14865:
14845:and a divisor
14834:
14810:
14791:
14777:
14776:
14774:
14771:
14770:
14769:
14764:
14758:
14753:
14748:
14743:
14741:Midy's theorem
14738:
14733:
14726:
14723:
14706: − 1
14692:
14691:
14678:
14674:
14668:
14664:
14658:
14654:
14650:
14647:
14644:
14641:
14638:
14600:
14597:
14558:
14557:
14546:
14541:
14538:
14534:
14531:
14528:
14525:
14522:
14519:
14516:
14513:
14508:
14503:
14500:
14497:
14494:
14491:
14477:
14476:
14465:
14462:
14459:
14456:
14453:
14450:
14447:
14444:
14440:
14437:
14431:
14426:
14423:
14420:
14417:
14414:
14411:
14408:
14394:
14393:
14382:
14377:
14373:
14370:
14362:
14356:
14351:
14347:
14344:
14338:
14334:
14331:
14326:
14321:
14318:
14313:
14309:
14306:
14286:floor function
13758:
13751:
13715:
13712:
13707:
13703:
13699:
13695:
13691:
13687:
13683:
13679:
13675:
13651:
13647:
13644:
13637:
13633:
13629:
13624:
13617:
13613:
13609:
13604:
13597:
13593:
13588:
13585:
13579:
13572:
13568:
13564:
13559:
13552:
13548:
13544:
13539:
13530:
13524:
13520:
13514:
13509:
13504:
13501:
13488:
13487:
13476:
13468:
13464:
13461:
13458:
13452:
13449:
13445:
13438:
13435:
13430:
13427:
13424:
13417:
13413:
13406:
13403:
13400:
13396:
13392:
13389:
13386:
13383:
13377:
13374:
13371:
13364:
13360:
13353:
13349:
13345:
13342:
13339:
13336:
13330:
13323:
13319:
13312:
13308:
13304:
13301:
13298:
13295:
13289:
13282:
13278:
13271:
13267:
13263:
13260:
13257:
13254:
13248:
13243:
13240:
13235:
13230:
13227:
13092:
13091:
13087:
13086:
13085:
13084:
13073:
13067:
13064:
13059:
13055:
13047:
13043:
13037:
13033:
13029:
13024:
13020:
13014:
13010:
13006:
12990:
12989:
12988:
12975:
12970:
12964:
12958:
12954:
12950:
12945:
12941:
12935:
12931:
12924:
12920:
12905:
12904:
12864:
12832:
12831:
12789:
12782:
12769:
12760:
12751:
12745:
12739:
12727:
12719:
12712:
12706:
12659:
12654:
12637:
12636:
12635:
12624:
12619:
12615:
12611:
12608:
12605:
12602:
12599:
12594:
12590:
12586:
12581:
12577:
12573:
12570:
12567:
12554:
12553:
12552:
12541:
12538:
12535:
12532:
12529:
12526:
12523:
12520:
12517:
12513:
12509:
12506:
12493:
12492:
12485:
12466:
12459:
12439:
12436:
12432:
12431:
12362:
12361:
12360:
12359:
12348:
12344:
12339:
12336:
12331:
12322:
12317:
12313:
12309:
12305:
12298:
12295:
12292:
12288:
12284:
12279:
12256:
12255:
12254:
12243:
12239:
12232:
12229:
12226:
12222:
12218:
12213:
12204:
12200:
12193:
12189:
12185:
12180:
12157:
12156:
12155:
12143:
12136:
12132:
12128:
12123:
12114:
12111:
12108:
12104:
12097:
12093:
12089:
12084:
12075:
12071:
12066:
12063:
12058:
12039:
12038:
12031:
11966:
11872:
11807:
11757:The period of
11755:
11726:
11697:
11670:The period of
11663:
11660:
11612:Main article:
11609:
11606:
11605:
11604:
11587:
11584:
11579:
11574:
11571:
11566:
11560:
11557:
11552:
11548:
11543:
11538:
11528:
11524:
11520:
11515:
11512:
11505:
11501:
11497:
11491:
11485:
11482:
11479:
11475:
11469:
11466:
11462:
11461:
11454:
11451:
11447:
11443:
11436:
11431:
11428:
11425:
11421:
11417:
11414:
11411:
11404:
11400:
11396:
11391:
11384:
11380:
11376:
11371:
11364:
11360:
11356:
11351:
11348:
11346:
11342:
11339:
11334:
11331:
11330:
11316:
11315:
11302:
11299:
11294:
11288:
11285:
11282:
11278:
11273:
11265:
11262:
11257:
11254:
11249:
11246:
11240:
11234:
11231:
11228:
11224:
11160:
11159:
11144:
11140:
11136:
11129:
11124:
11121:
11118:
11114:
11110:
11107:
11104:
11099:
11096:
11091:
11086:
11083:
11078:
11073:
11070:
11065:
11060:
11057:
11052:
11033:
11030:
11017:
11014:
11010:
11006:
10985:
10964:
10935:
10932:
10926:
10923:
10918:
10915:
10909:
10906:
10901:
10897:
10894:
10891:
10884:
10881:
10877:
10871:
10868:
10864:
10860:
10857:
10855:
10852:
10848:
10844:
10841:
10839:
10836:
10833:
10829:
10825:
10823:
10820:
10816:
10812:
10810:
10807:
10803:
10799:
10798:
10796:
10791:
10788:
10784:
10781:
10776:
10773:
10770:
10768:
10765:
10762:
10761:
10758:
10752:
10749:
10743:
10740:
10735:
10732:
10726:
10723:
10718:
10714:
10711:
10708:
10701:
10698:
10694:
10688:
10685:
10681:
10677:
10674:
10672:
10669:
10665:
10661:
10658:
10656:
10653:
10650:
10646:
10642:
10640:
10637:
10633:
10629:
10627:
10624:
10620:
10616:
10615:
10613:
10608:
10605:
10601:
10598:
10593:
10590:
10587:
10585:
10582:
10579:
10578:
10575:
10569:
10566:
10560:
10557:
10552:
10548:
10545:
10542:
10535:
10532:
10528:
10522:
10519:
10515:
10511:
10508:
10506:
10503:
10499:
10495:
10492:
10490:
10487:
10484:
10480:
10476:
10474:
10471:
10467:
10463:
10461:
10458:
10454:
10450:
10449:
10447:
10442:
10439:
10435:
10432:
10427:
10424:
10421:
10419:
10416:
10413:
10412:
10409:
10403:
10400:
10394:
10391:
10386:
10382:
10379:
10376:
10369:
10366:
10362:
10356:
10353:
10349:
10345:
10342:
10340:
10337:
10333:
10329:
10326:
10324:
10321:
10318:
10314:
10310:
10308:
10305:
10301:
10297:
10295:
10292:
10288:
10284:
10283:
10281:
10276:
10273:
10269:
10266:
10261:
10258:
10255:
10253:
10250:
10247:
10246:
10243:
10237:
10234:
10228:
10225:
10220:
10216:
10213:
10210:
10203:
10200:
10196:
10190:
10187:
10183:
10179:
10176:
10174:
10171:
10167:
10163:
10160:
10158:
10155:
10152:
10148:
10144:
10142:
10139:
10135:
10131:
10129:
10126:
10122:
10118:
10117:
10115:
10110:
10107:
10103:
10100:
10095:
10092:
10089:
10087:
10084:
10081:
10080:
10054:
10023:
10019:
9999:
9978:
9974:
9954:
9941:Formation rule
9922:
9918:
9897:
9875:
9871:
9850:
9828:
9815:
9812:
9719:
9718:
9683:
9644:
9643:
9642:
9641:
9461:
9460:
9459:
9290:
9289:
9254:
9237:0.005656... =
9235:
9218:0.000444... =
9196:
9195:
9178:0.999999... =
9176:
9143:0.012012... =
9141:
9124:0.565656... =
9122:
9105:0.444444... =
9084:
9083:
9065:
9062:
9059:
9052:
9048:
9044:
9039:
9035:
9029:
9025:
9018:
9012:
9009:
9004:
9000:
8992:
8988:
8984:
8979:
8975:
8969:
8965:
8958:
8955:
8953:
8951:
8948:
8947:
8942:
8938:
8934:
8929:
8925:
8919:
8915:
8911:
8908:
8906:
8904:
8901:
8898:
8895:
8892:
8889:
8885:
8881:
8878:
8873:
8869:
8864:
8860:
8859:
8854:
8848:
8844:
8840:
8835:
8831:
8825:
8821:
8814:
8809:
8805:
8801:
8796:
8792:
8786:
8782:
8778:
8775:
8773:
8771:
8766:
8762:
8758:
8757:
8752:
8746:
8742:
8738:
8733:
8729:
8723:
8719:
8712:
8709:
8706:
8704:
8702:
8699:
8698:
8680:
8679:
8662:
8659:
8654:
8649:
8645:
8642:
8639:
8636:
8633:
8627:
8622:
8619:
8614:
8611:
8608:
8605:
8603:
8601:
8598:
8595:
8592:
8589:
8586:
8583:
8581:
8579:
8576:
8573:
8572:
8558:
8557:
8540:
8537:
8532:
8526:
8523:
8520:
8515:
8512:
8509:
8506:
8503:
8500:
8497:
8491:
8488:
8486:
8484:
8479:
8476:
8471:
8466:
8463:
8458:
8452:
8449:
8446:
8441:
8438:
8435:
8429:
8424:
8421:
8416:
8411:
8408:
8403:
8398:
8395:
8390:
8387:
8385:
8383:
8380:
8377:
8374:
8371:
8368:
8365:
8363:
8361:
8358:
8355:
8354:
8320:
8319:
8302:
8299:
8294:
8288:
8285:
8280:
8276:
8271:
8266:
8263:
8261:
8259:
8256:
8255:
8252:
8249:
8246:
8244:
8242:
8239:
8236:
8233:
8229:
8225:
8222:
8217:
8213:
8208:
8204:
8203:
8200:
8197:
8194:
8191:
8189:
8187:
8182:
8178:
8174:
8173:
8170:
8167:
8164:
8161:
8159:
8157:
8154:
8153:
8130:
8127:
8126:
8125:
8122:
8121:
8118:
8105:
8102:
8097:
8091:
8088:
8085:
8080:
8077:
8074:
8068:
8063:
8060:
8055:
8045:
8034:
8023:
8022:
8019:
8008:
8005:
7995:
7984:
7981:
7970:
7969:
7966:
7955:
7952:
7949:
7939:
7928:
7925:
7914:
7913:
7910:
7899:
7896:
7881:
7871:
7860:
7857:
7846:
7845:
7834:
7831:
7816:
7806:
7795:
7777:
7776:
7773:
7772:
7769:
7756:
7753:
7748:
7743:
7740:
7735:
7725:
7714:
7703:
7702:
7699:
7688:
7685:
7675:
7664:
7661:
7650:
7649:
7646:
7635:
7632:
7629:
7619:
7608:
7605:
7594:
7593:
7582:
7579:
7576:
7566:
7555:
7536:
7533:
7532:
7531:
7512:
7509:
7466:
7465:
7440:
7414:
7413:
7402:
7395:
7390:
7386:
7380:
7376:
7370:
7366:
7361:
7337:
7336:
7325:
7318:
7313:
7309:
7303:
7299:
7293:
7289:
7282:
7279:
7276:
7272:
7247:
7246:
7235:
7228:
7223:
7219:
7213:
7209:
7203:
7199:
7192:
7189:
7186:
7182:
7154:
7153:
7142:
7135:
7130:
7126:
7120:
7116:
7110:
7106:
7102:
7097:
7093:
7088:
7070:
7067:
7007:
7000:
6993:
6988:
6987:
6976:
6973:
6970:
6963:
6959:
6954:
6950:
6943:
6939:
6934:
6930:
6923:
6919:
6914:
6910:
6907:
6904:
6890:
6889:
6875:
6870:
6866:
6860:
6856:
6850:
6846:
6841:
6777:
6750:
6741:
6732:
6704:
6703:
6640:
6639:
6621:
6564:
6463:
6454:
6432:The period of
6415:
6414:
6402:
6399:
6395:
6392:
6387:
6384:
6379:
6375:
6320:
6319:
6268:
6265:
6214:if and only if
6128:
6125:
6098:
6097:
6096:= 0.615384...,
6079:
6061:
6043:
6025:
6007:
5985:
5984:
5983:= 0.307692...,
5966:
5948:
5930:
5912:
5894:
5853:
5852:
5841:
5838:
5833:
5829:
5826:
5821:
5818:
5815:
5811:
5732:
5731:
5709:
5687:
5665:
5643:
5621:
5599:
5577:
5555:
5533:
5511:
5489:
5467:
5445:
5418:
5415:
5414:
5413:
5347:
5346:
5320: − 1
5224:
5223:
5198:
5173:
5148:
5123:
5098:
4892:
4891:
4869:
4847:
4825:
4803:
4781:
4759:
4737:
4715:
4685:Main article:
4682:
4681:Cyclic numbers
4679:
4611:primitive root
4565:Main article:
4562:
4559:
4558:
4557:
4495:
4494:
4446:
4445:
4401:
4400:
4360:
4359:
4313:
4307:
4306:
4261:
4251:
4219:
4218:
4215:
4214:
4211:
4205:
4187:
4186:
4183:
4177:
4159:
4158:
4155:
4149:
4131:
4130:
4127:
4121:
4103:
4102:
4099:
4093:
4075:
4074:
4071:
4065:
4047:
4046:
4043:
4037:
4019:
4018:
4015:
4012:
3994:
3993:
3990:
3984:
3966:
3965:
3962:
3956:
3938:
3937:
3934:
3928:
3910:
3909:
3906:
3900:
3882:
3881:
3878:
3872:
3854:
3853:
3850:
3844:
3826:
3825:
3822:
3816:
3798:
3797:
3794:
3791:
3773:
3772:
3768:
3762:
3757:
3752:
3746:
3743:
3742:
3739:
3733:
3715:
3714:
3711:
3705:
3687:
3686:
3683:
3677:
3659:
3658:
3655:
3649:
3631:
3630:
3627:
3621:
3603:
3602:
3599:
3593:
3575:
3574:
3571:
3568:
3550:
3549:
3546:
3540:
3522:
3521:
3518:
3512:
3494:
3493:
3490:
3484:
3466:
3465:
3462:
3456:
3438:
3437:
3434:
3431:
3413:
3412:
3409:
3403:
3385:
3384:
3381:
3375:
3357:
3356:
3353:
3347:
3329:
3328:
3324:
3318:
3313:
3308:
3302:
3299:
3298:
3295:
3292:
3289:
3286:
3268:
3267:
3264:
3258:
3255:
3249:
3231:
3230:
3227:
3221:
3218:
3212:
3194:
3193:
3190:
3184:
3181:
3175:
3157:
3156:
3153:
3147:
3144:
3138:
3120:
3119:
3116:
3110:
3107:
3101:
3083:
3082:
3079:
3073:
3070:
3067:
3049:
3048:
3045:
3039:
3036:
3030:
3012:
3011:
3008:
3005:
3002:
2999:
2981:
2980:
2977:
2971:
2968:
2962:
2944:
2943:
2940:
2934:
2931:
2925:
2907:
2906:
2903:
2897:
2894:
2891:
2873:
2872:
2869:
2866:
2863:
2860:
2842:
2841:
2838:
2832:
2829:
2823:
2805:
2804:
2801:
2798:
2795:
2792:
2774:
2773:
2769:
2763:
2758:
2754:
2748:
2743:
2738:
2731:
2729:
2726:
2713:
2690:
2687:
2682:
2678:
2672:
2667:
2663:
2639:
2636:
2633:
2613:
2591:
2585:
2582:
2577:
2573:
2567:
2562:
2558:
2551:
2548:
2545:
2542:
2539:
2536:
2508:
2505:
2501:
2497:
2494:
2490:
2485:
2478:
2473:
2468:
2464:
2460:
2455:
2448:
2443:
2440:
2437:
2433:
2409:
2406:
2403:
2399:
2391:
2386:
2381:
2377:
2373:
2368:
2361:
2356:
2353:
2350:
2346:
2342:
2338:
2334:
2331:
2328:
2321:
2316:
2311:
2307:
2303:
2298:
2291:
2286:
2283:
2280:
2276:
2272:
2269:
2266:
2263:
2231:
2227:
2202:
2198:
2194:
2191:
2188:
2182:
2178:
2172:
2168:
2164:
2161:
2138:
2134:
2131:
2108:
2105:
2099:
2096:
2093:
2090:
2069:
2065:
2062:
2042:
2039:
2036:
2016:
2013:
2010:
1988:
1982:
1979:
1973:
1970:
1967:
1964:
1961:
1956:
1952:
1927:
1923:
1902:
1882:
1878:
1875:
1870:
1866:
1861:
1858:
1855:
1835:
1815:
1795:
1773:
1770:
1765:
1762:
1759:
1756:
1753:
1741:
1738:
1730:
1729:
1726:
1725:
1691:
1684:
1683:
1680:
1673:
1672:
1669:
1639:
1636:
1611:
1608:
1549:560
1538:
1509:
1506:
1492:
1491:
1475:
1460:Latin American
1449:
1400:: In parts of
1395:
1360:United Kingdom
1325:
1306:Czech Republic
1268:
1267:
1258:
1252:
1249:
1225:
1219:
1201:
1183:
1182:
1173:
1167:
1164:
1140:
1134:
1116:
1098:
1097:
1088:
1082:
1081:0.(012345679)
1079:
1055:
1049:
1031:
1013:
1012:
1003:
997:
994:
970:
964:
946:
928:
927:
918:
912:
909:
896:
890:
872:
854:
853:
850:
844:
841:
818:
812:
794:
776:
775:
772:
766:
763:
750:
744:
726:
708:
707:
704:
698:
695:
682:
676:
658:
640:
639:
636:
630:
627:
614:
608:
606:
588:
587:
582:
579:
574:
571:
566:
552:
551:
502:
500:
493:
487:
484:
482:
479:
452:is said to be
245:
244:
227:
226:
181:
179:
172:
165:
164:
79:
77:
70:
65:
39:
38:
36:
29:
15:
9:
6:
4:
3:
2:
15383:
15372:
15369:
15367:
15364:
15363:
15361:
15349:
15348:
15343:
15340:
15335:
15334:
15323:
15319:
15313:
15306:
15300:
15293:
15287:
15280:
15274:
15267:
15265:
15260:
15255:
15248:
15247:
15240:
15233:
15227:
15218:
15211:
15205:
15198:
15194:
15190:
15186:
15182:
15178:
15174:
15170:
15166:
15165:
15157:
15142:
15138:
15134:
15127:
15112:
15108:
15104:
15098:
15084:
15080:
15073:
15066:
15062:
15059:
15038:
15035:
15029:
15026:
15023:
15013:
15006:
15000:
14995:
14991:
14981:
14975:
14969:
14962:
14961:
14958:
14953:
14930:
14922:
14919:
14914:
14910:
14905:
14896:
14893:
14884:
14878:
14872:
14867:
14863:
14855:
14854:
14852:
14848:
14844:
14838:
14824:
14820:
14814:
14806:
14802:
14795:
14788:
14782:
14778:
14768:
14765:
14762:
14759:
14757:
14754:
14752:
14751:Trailing zero
14749:
14747:
14744:
14742:
14739:
14737:
14734:
14732:
14729:
14728:
14722:
14720:
14719:diehard tests
14705:
14697:
14676:
14666:
14656:
14652:
14648:
14642:
14636:
14629:
14628:
14627:
14625:
14617:
14596:
14594:
14575:
14544:
14539:
14536:
14532:
14529:
14526:
14523:
14520:
14517:
14514:
14511:
14501:
14498:
14495:
14492:
14489:
14482:
14481:
14480:
14463:
14460:
14457:
14454:
14451:
14448:
14445:
14442:
14438:
14435:
14424:
14421:
14418:
14415:
14412:
14409:
14406:
14399:
14398:
14397:
14380:
14375:
14371:
14368:
14360:
14354:
14349:
14345:
14342:
14336:
14332:
14329:
14324:
14319:
14316:
14311:
14307:
14304:
14294:
14293:
14292:
14287:
14279:
14270:
13756:
13750:
13746:
13735:
13727:
13711:
13669:
13649:
13645:
13642:
13635:
13631:
13627:
13622:
13615:
13611:
13607:
13602:
13595:
13591:
13586:
13583:
13577:
13570:
13566:
13562:
13557:
13550:
13546:
13542:
13537:
13528:
13522:
13518:
13512:
13507:
13502:
13499:
13474:
13466:
13462:
13459:
13456:
13450:
13447:
13443:
13436:
13433:
13428:
13425:
13422:
13415:
13411:
13404:
13401:
13398:
13390:
13387:
13384:
13375:
13372:
13369:
13362:
13358:
13351:
13343:
13340:
13337:
13328:
13321:
13317:
13310:
13302:
13299:
13296:
13287:
13280:
13276:
13269:
13261:
13258:
13255:
13246:
13241:
13238:
13233:
13228:
13225:
13216:
13215:
13214:
13203:
13097:
13089:
13088:
13071:
13065:
13062:
13057:
13053:
13045:
13035:
13031:
13027:
13022:
13018:
13012:
13008:
12994:
12993:
12991:
12973:
12968:
12956:
12952:
12948:
12943:
12939:
12933:
12929:
12922:
12918:
12909:
12908:
12907:
12906:
12896:
12892:
12888:
12883:
12879:
12872:
12867:
12861:
12855:
12834:
12833:
12814: =
12805:
12792:
12785:
12776:
12772:
12759:
12754:
12744:
12738:
12730:
12722:
12718:
12711:
12705:
12698:
12694:
12690:
12686:
12682:
12678:
12673:
12665:
12657:
12643:
12638:
12617:
12613:
12609:
12606:
12603:
12600:
12597:
12592:
12588:
12584:
12579:
12575:
12568:
12565:
12558:
12557:
12555:
12536:
12533:
12530:
12527:
12524:
12521:
12515:
12507:
12504:
12497:
12496:
12495:
12494:
12490:
12486:
12482:
12476:
12471:
12467:
12464:
12460:
12457:
12453:
12452:decimal point
12449:
12445:
12444:
12443:
12435:
12418:
12410:
12406:
12402:
12398:
12394:
12386:
12372:
12368:
12364:
12363:
12346:
12342:
12337:
12334:
12329:
12320:
12315:
12311:
12307:
12303:
12296:
12293:
12290:
12286:
12282:
12277:
12264:
12263:
12261:
12257:
12241:
12237:
12230:
12227:
12224:
12220:
12216:
12211:
12202:
12198:
12191:
12187:
12183:
12178:
12165:
12164:
12162:
12158:
12141:
12134:
12130:
12126:
12121:
12112:
12109:
12106:
12102:
12095:
12091:
12087:
12082:
12073:
12069:
12064:
12061:
12056:
12043:
12042:
12041:
12040:
12036:
12032:
12025:
12007:
11989:
11975:
11971:
11967:
11960:
11942:
11924:
11905:
11902:
11899:
11885:
11881:
11877:
11873:
11868:
11862:
11858:
11854:
11850:
11846:
11838:
11824:
11820:
11816:
11812:
11808:
11801:
11787:
11783:
11780:
11772:
11764:
11756:
11753:
11745:
11731:
11727:
11724:
11716:
11702:
11698:
11695:
11691:
11683:
11669:
11668:
11667:
11659:
11621:
11615:
11585:
11582:
11577:
11572:
11569:
11564:
11558:
11555:
11550:
11546:
11541:
11536:
11526:
11522:
11518:
11513:
11510:
11503:
11499:
11495:
11489:
11483:
11480:
11477:
11473:
11452:
11449:
11445:
11441:
11429:
11426:
11423:
11419:
11415:
11412:
11409:
11402:
11398:
11394:
11389:
11382:
11378:
11374:
11369:
11362:
11358:
11354:
11349:
11347:
11337:
11332:
11321:
11320:
11319:
11300:
11297:
11292:
11286:
11283:
11280:
11276:
11271:
11263:
11260:
11255:
11252:
11247:
11244:
11238:
11232:
11229:
11226:
11222:
11213:
11212:
11211:
11209:
11205:
11201:
11165:
11142:
11138:
11134:
11122:
11119:
11116:
11112:
11108:
11105:
11102:
11097:
11094:
11089:
11084:
11081:
11076:
11071:
11068:
11063:
11055:
11050:
11043:
11042:
11041:
11039:
11029:
11015:
11012:
10953:
10933:
10930:
10924:
10916:
10913:
10907:
10899:
10895:
10892:
10889:
10882:
10875:
10869:
10866:
10853:
10850:
10834:
10831:
10821:
10818:
10808:
10805:
10794:
10789:
10779:
10774:
10771:
10766:
10763:
10750:
10747:
10741:
10733:
10730:
10724:
10716:
10712:
10709:
10706:
10699:
10692:
10686:
10683:
10670:
10667:
10651:
10648:
10635:
10625:
10622:
10611:
10606:
10596:
10591:
10588:
10583:
10580:
10567:
10564:
10558:
10550:
10546:
10543:
10540:
10533:
10526:
10520:
10517:
10504:
10501:
10485:
10482:
10472:
10469:
10459:
10456:
10445:
10440:
10430:
10425:
10422:
10417:
10414:
10401:
10398:
10392:
10384:
10380:
10377:
10374:
10367:
10360:
10354:
10351:
10338:
10335:
10319:
10316:
10303:
10293:
10290:
10279:
10274:
10264:
10259:
10256:
10251:
10248:
10235:
10232:
10226:
10218:
10214:
10211:
10208:
10201:
10194:
10188:
10185:
10172:
10169:
10153:
10150:
10140:
10137:
10127:
10124:
10113:
10108:
10098:
10093:
10090:
10085:
10082:
10070:
10067:
10043:
10042:
10036:
9997:
9952:
9939:
9935:
9811:
9793:
9789:
9785:
9764:
9762:
9758:
9750:
9742:
9733:
9731:
9727:
9724:
9684:
9650:1.23444... =
9649:
9648:
9647:
9577:+ 978/999 = −
9560:
9559:
9462:
9378:
9377:
9296:
9295:
9294:
9255:
9236:
9217:
9216:
9215:
9213:
9209:
9205:
9201:
9177:
9142:
9123:
9104:
9103:
9102:
9100:
9096:
9092:
9087:
9063:
9060:
9057:
9050:
9046:
9042:
9037:
9033:
9027:
9023:
9016:
9010:
9007:
9002:
8998:
8990:
8986:
8982:
8977:
8973:
8967:
8963:
8956:
8954:
8949:
8940:
8936:
8932:
8927:
8923:
8917:
8913:
8909:
8907:
8902:
8899:
8896:
8893:
8890:
8887:
8883:
8879:
8876:
8871:
8867:
8862:
8846:
8842:
8838:
8833:
8829:
8823:
8819:
8812:
8807:
8803:
8799:
8794:
8790:
8784:
8780:
8776:
8774:
8769:
8764:
8760:
8744:
8740:
8736:
8731:
8727:
8721:
8717:
8710:
8707:
8705:
8700:
8689:
8688:
8687:
8685:
8660:
8657:
8652:
8647:
8643:
8640:
8637:
8634:
8631:
8625:
8620:
8617:
8612:
8609:
8606:
8604:
8596:
8593:
8590:
8587:
8584:
8582:
8577:
8574:
8563:
8562:
8561:
8538:
8535:
8530:
8524:
8521:
8518:
8513:
8510:
8507:
8504:
8501:
8498:
8495:
8489:
8487:
8477:
8474:
8469:
8464:
8461:
8456:
8450:
8447:
8444:
8439:
8436:
8433:
8427:
8422:
8419:
8414:
8409:
8406:
8401:
8396:
8393:
8388:
8386:
8378:
8375:
8372:
8369:
8366:
8364:
8359:
8356:
8345:
8344:
8343:
8341:
8300:
8297:
8292:
8286:
8283:
8278:
8274:
8269:
8264:
8262:
8257:
8250:
8247:
8245:
8240:
8237:
8234:
8231:
8227:
8223:
8220:
8215:
8211:
8206:
8198:
8195:
8192:
8190:
8185:
8180:
8176:
8168:
8165:
8162:
8160:
8155:
8144:
8143:
8142:
8140:
8136:
8119:
8103:
8100:
8095:
8089:
8086:
8083:
8078:
8075:
8072:
8066:
8061:
8058:
8053:
8046:
8032:
8025:
8024:
8020:
8006:
8003:
7996:
7982:
7979:
7972:
7971:
7967:
7953:
7950:
7947:
7940:
7926:
7923:
7916:
7915:
7911:
7897:
7894:
7879:
7872:
7858:
7855:
7848:
7847:
7832:
7829:
7814:
7807:
7793:
7786:
7785:
7782:
7781:
7780:
7770:
7754:
7751:
7746:
7741:
7738:
7733:
7726:
7712:
7705:
7704:
7700:
7686:
7683:
7676:
7662:
7659:
7652:
7651:
7647:
7633:
7630:
7627:
7620:
7606:
7603:
7596:
7595:
7580:
7577:
7574:
7567:
7553:
7546:
7545:
7542:
7541:
7540:
7513:
7510:
7506:
7503:
7498:
7494:
7491:
7490:
7489:
7458:
7455:
7441:
7438:
7434:
7430:
7429:
7428:
7425:
7423:
7419:
7400:
7393:
7388:
7384:
7378:
7374:
7368:
7364:
7359:
7350:
7349:
7348:
7346:
7342:
7323:
7316:
7311:
7307:
7301:
7297:
7291:
7287:
7280:
7277:
7274:
7270:
7260:
7259:
7258:
7256:
7252:
7233:
7226:
7221:
7217:
7211:
7207:
7201:
7197:
7190:
7187:
7184:
7180:
7170:
7169:
7168:
7165:
7163:
7159:
7140:
7133:
7128:
7124:
7118:
7114:
7108:
7104:
7100:
7095:
7091:
7086:
7077:
7076:
7075:
7066:
7060:
7042:
7024:
7010:
7003:
6996:
6971:
6968:
6961:
6957:
6952:
6948:
6941:
6937:
6932:
6928:
6921:
6917:
6912:
6905:
6902:
6895:
6894:
6893:
6873:
6868:
6864:
6858:
6854:
6848:
6844:
6839:
6830:
6829:
6828:
6826:
6822:
6818:
6814:
6810:
6806:
6801:
6795:
6780:
6776:
6768:
6753:
6749:
6744:
6740:
6735:
6731:
6723:
6709:
6682:
6681:
6680:
6678:
6674:
6666:
6652:
6647:
6645:
6637:
6633:
6629:
6625:
6622:
6619:
6616:
6615:
6614:
6592:
6578:
6574:
6569:
6567:
6563:
6560:
6552:
6537:
6535:
6530:
6525:
6517:
6499:
6481:
6466:
6462:
6457:
6453:
6445:
6430:
6428:
6424:
6420:
6397:
6393:
6385:
6382:
6377:
6373:
6365:
6364:
6363:
6361:
6357:
6353:
6349:
6345:
6341:
6337:
6333:
6329:
6325:
6309:
6296:
6295:
6294:
6288:
6274:
6264:
6262:
6258:
6254:
6250:
6247:= 1, 2, ...,
6246:
6238:
6230:
6222:
6218:
6215:
6210:
6206:
6202:
6196:
6194:
6193:
6185:
6181:
6176:
6172:
6168:
6164:
6156:
6142:
6138:
6135:, the length
6134:
6124:
6122:
6119: =
6118:
6114:
6110:
6106:
6101:
6080:
6078:= 0.461538...
6062:
6060:= 0.846153...
6044:
6042:= 0.384615...
6026:
6024:= 0.538461...
6008:
6006:= 0.153846...
5990:
5989:
5988:
5967:
5965:= 0.230769...
5949:
5947:= 0.923076...
5931:
5929:= 0.692307...
5913:
5911:= 0.769230...
5895:
5893:= 0.076923...
5877:
5876:
5875:
5856:
5839:
5836:
5831:
5827:
5824:
5819:
5816:
5813:
5809:
5798:
5797:
5796:
5790:
5776:
5772:
5768:
5760:
5745:
5743:
5738:
5710:
5688:
5684:0126582278481
5666:
5644:
5622:
5600:
5596:0188679245283
5578:
5556:
5534:
5512:
5490:
5468:
5446:
5424:
5423:
5422:
5411:
5406:
5401:
5400:
5399:
5397:
5393:
5385:
5371:
5367:
5363:
5358:
5357:to 1 mod 10.
5356:
5352:
5344:
5339:
5334:
5333:
5332:
5319:
5311:
5307:
5303:
5298:
5294:
5292:
5256:
5251:
5249:
5199:
5174:
5149:
5124:
5099:
5074:
5073:
5072:
5070:
5065:
5063:
5058:
4870:
4848:
4826:
4804:
4782:
4760:
4738:
4716:
4694:
4693:
4692:
4688:
4687:Cyclic number
4678:
4676:
4675:cyclic number
4672:
4668:
4660:
4645:
4643:
4638:
4634:
4628:
4624:
4620:
4616:
4612:
4609:. If 10 is a
4608:
4605:of 10 modulo
4604:
4596:
4582:
4578:
4574:
4568:
4555:
4550:
4544:
4543:
4542:
4540:
4534:
4530:
4524:
4516:
4508:
4500:
4492:
4487:
4481:
4480:
4479:
4477:
4473:
4465:
4451:
4443:
4438:
4432:
4431:
4430:
4428:
4424:
4416:
4398:
4393:
4387:
4386:
4385:
4383:
4375:
4357:
4353:
4349:
4348:
4347:
4345:
4337:
4323:
4319:
4312:
4304:
4299:
4293:
4292:
4291:
4289:
4281:
4267:
4260:
4255:
4250:
4242:
4229:
4228:unit fraction
4225:
4212:
4206:
4189:
4188:
4184:
4178:
4161:
4160:
4156:
4150:
4133:
4132:
4128:
4122:
4105:
4104:
4100:
4094:
4077:
4076:
4072:
4066:
4049:
4048:
4044:
4038:
4021:
4020:
4016:
4013:
3996:
3995:
3991:
3985:
3968:
3967:
3963:
3957:
3940:
3939:
3935:
3929:
3912:
3911:
3907:
3901:
3884:
3883:
3879:
3873:
3856:
3855:
3851:
3845:
3828:
3827:
3823:
3817:
3800:
3799:
3795:
3792:
3775:
3774:
3767:
3763:
3758:
3755:
3751:
3750:
3747:
3740:
3734:
3717:
3716:
3712:
3706:
3689:
3688:
3684:
3678:
3661:
3660:
3656:
3650:
3633:
3632:
3628:
3622:
3605:
3604:
3600:
3594:
3577:
3576:
3572:
3569:
3552:
3551:
3547:
3541:
3524:
3523:
3519:
3513:
3496:
3495:
3491:
3485:
3468:
3467:
3463:
3457:
3440:
3439:
3435:
3432:
3415:
3414:
3410:
3404:
3387:
3386:
3382:
3376:
3359:
3358:
3354:
3348:
3331:
3330:
3323:
3319:
3314:
3311:
3307:
3306:
3303:
3296:
3293:
3290:
3287:
3270:
3269:
3265:
3259:
3256:
3250:
3233:
3232:
3228:
3222:
3219:
3213:
3196:
3195:
3191:
3185:
3182:
3176:
3159:
3158:
3154:
3148:
3145:
3139:
3122:
3121:
3117:
3111:
3108:
3102:
3085:
3084:
3080:
3074:
3071:
3068:
3051:
3050:
3046:
3040:
3037:
3031:
3014:
3013:
3009:
3006:
3003:
3000:
2983:
2982:
2978:
2972:
2969:
2963:
2946:
2945:
2941:
2935:
2932:
2926:
2909:
2908:
2904:
2898:
2895:
2892:
2875:
2874:
2870:
2867:
2864:
2861:
2844:
2843:
2839:
2833:
2830:
2824:
2807:
2806:
2802:
2799:
2796:
2793:
2776:
2775:
2768:
2764:
2759:
2753:
2749:
2744:
2741:
2737:
2736:
2733:
2732:
2725:
2711:
2688:
2685:
2680:
2676:
2670:
2665:
2661:
2637:
2634:
2631:
2611:
2602:
2589:
2583:
2580:
2575:
2571:
2565:
2560:
2556:
2549:
2546:
2543:
2540:
2537:
2534:
2526:
2506:
2503:
2499:
2495:
2492:
2488:
2483:
2476:
2466:
2462:
2453:
2441:
2438:
2435:
2431:
2420:
2407:
2404:
2401:
2397:
2389:
2379:
2375:
2366:
2354:
2351:
2348:
2344:
2340:
2336:
2332:
2329:
2326:
2319:
2309:
2305:
2296:
2284:
2281:
2278:
2274:
2270:
2267:
2264:
2261:
2253:
2251:
2247:
2229:
2225:
2215:
2200:
2196:
2192:
2189:
2186:
2180:
2176:
2170:
2166:
2162:
2159:
2151:
2132:
2129:
2103:
2097:
2094:
2091:
2088:
2063:
2060:
2040:
2037:
2034:
2014:
2011:
2008:
1999:
1986:
1977:
1971:
1968:
1965:
1962:
1959:
1954:
1950:
1941:
1925:
1921:
1900:
1876:
1873:
1868:
1864:
1856:
1853:
1833:
1813:
1793:
1768:
1763:
1760:
1757:
1754:
1751:
1737:
1735:
1692:
1690:
1685:
1681:
1679:
1674:
1668:
1664:
1659:
1656:
1655:
1654:
1650:
1645:
1635:
1633:
1629:
1625:
1620:
1616:
1607:
1605:
1602:× 10 (modulo
1601:
1597:
1589:
1581:
1572:
1556:
1552:
1548:
1544:
1537:
1519:
1518:long division
1515:
1505:
1501:1886792452830
1489:
1483:
1479:
1476:
1473:
1469:
1465:
1461:
1457:
1453:
1450:
1447:
1443:
1439:
1436:, as well as
1435:
1431:
1427:
1423:
1419:
1415:
1411:
1407:
1403:
1399:
1396:
1393:
1389:
1385:
1381:
1377:
1373:
1369:
1365:
1361:
1357:
1353:
1349:
1345:
1341:
1337:
1333:
1329:
1326:
1323:
1319:
1315:
1311:
1307:
1303:
1299:
1295:
1291:
1287:
1283:
1279:
1278:United States
1275:
1272:
1271:
1263:1886792452830
1259:
1256:1886792452830
1253:
1250:
1226:
1223:1886792452830
1220:
1214:9999999999999
1202:
1185:
1184:
1174:
1168:
1165:
1141:
1135:
1117:
1100:
1099:
1089:
1083:
1080:
1056:
1050:
1032:
1015:
1014:
1004:
998:
995:
971:
965:
947:
930:
929:
919:
913:
910:
897:
891:
873:
856:
855:
851:
845:
842:
819:
813:
795:
778:
777:
773:
767:
764:
751:
745:
727:
710:
709:
705:
699:
696:
683:
677:
659:
642:
641:
637:
631:
628:
615:
609:
607:
590:
589:
586:
583:
580:
578:
575:
572:
570:
567:
563:
557:
548:
545:
537:
527:
523:
519:
513:
512:
508:
503:This section
501:
497:
492:
491:
478:
476:
470:
459:
455:
451:
447:
442:
440:
432:
426:
422:
389:
381:
358:
354:
350:
346:
342:
337:
319:
299:
298:decimal point
279:
274:
272:
268:
264:
260:
256:
252:
241:
238:
223:
220:
212:
202:
197:
192:
188:
187:
180:
171:
170:
161:
158:
150:
139:
136:
132:
129:
125:
122:
118:
115:
111:
108: –
107:
103:
102:Find sources:
96:
92:
86:
85:
80:This article
78:
74:
69:
68:
63:
61:
54:
53:
48:
47:
42:
37:
28:
27:
22:
15345:
15312:
15304:
15299:
15291:
15286:
15278:
15273:
15262:
15254:
15244:
15239:
15234:, p. 79
15231:
15226:
15217:
15209:
15204:
15196:
15168:
15162:
15156:
15145:. Retrieved
15141:the original
15136:
15126:
15115:. Retrieved
15111:the original
15107:www.sjsu.edu
15106:
15097:
15086:. Retrieved
15082:
15072:
15064:
15060:
14956:
14846:
14842:
14837:
14826:. Retrieved
14822:
14813:
14804:
14800:
14794:
14786:
14781:
14756:Unique prime
14703:
14695:
14693:
14623:
14615:
14602:
14559:
14478:
14395:
14271:
14264:
13814:""
13748:
13744:
13733:
13725:
13717:
13670:
13489:
13204:
13093:
12894:
12890:
12881:
12877:
12870:
12865:
12858:. It is the
12853:
12787:
12780:
12774:
12767:
12757:
12749:
12742:
12736:
12725:
12720:
12716:
12709:
12703:
12696:
12676:
12671:
12663:
12652:
12641:
12480:
12474:
12462:
12441:
12433:
12416:
12408:
12404:
12400:
12396:
12392:
12384:
12371:proper prime
12370:
12366:
12259:
12160:
12034:
12023:
12005:
11987:
11973:
11969:
11958:
11940:
11922:
11903:
11900:
11897:
11883:
11879:
11875:
11866:
11865:10 ≡ 1 (mod
11856:
11852:
11848:
11844:
11836:
11822:
11818:
11814:
11810:
11799:
11785:
11778:
11770:
11762:
11751:
11743:
11729:
11722:
11714:
11700:
11693:
11689:
11688:for integer
11681:
11665:
11617:
11317:
11207:
11203:
11161:
11035:
10954:
10071:
10068:
10040:
10039:
10037:
9944:
9817:
9791:
9787:
9783:
9765:
9760:
9756:
9748:
9740:
9734:
9729:
9725:
9720:
9645:
9291:
9211:
9207:
9203:
9199:
9197:
9098:
9094:
9090:
9088:
9085:
8683:
8681:
8559:
8339:
8321:
8138:
8134:
8132:
7951:836.36363636
7778:
7538:
7504:
7501:
7496:
7492:
7468:For example
7467:
7456:
7453:
7436:
7432:
7426:
7421:
7417:
7415:
7344:
7340:
7338:
7254:
7250:
7248:
7166:
7161:
7157:
7155:
7072:
7058:
7040:
7022:
7005:
6998:
6991:
6989:
6891:
6824:
6820:
6816:
6812:
6808:
6804:
6802:
6793:
6778:
6774:
6766:
6751:
6747:
6742:
6738:
6733:
6729:
6721:
6707:
6705:
6676:
6672:
6664:
6650:
6648:
6641:
6635:
6631:
6623:
6617:
6590:
6576:
6572:
6570:
6565:
6561:
6558:
6550:
6538:
6523:
6515:
6497:
6479:
6464:
6460:
6455:
6451:
6443:
6431:
6426:
6418:
6416:
6359:
6355:
6351:
6347:
6335:
6331:
6327:
6323:
6321:
6307:
6286:
6272:
6270:
6256:
6252:
6248:
6244:
6236:
6228:
6220:
6216:
6208:
6204:
6200:
6197:
6191:
6183:
6179:
6170:
6166:
6162:
6154:
6140:
6136:
6132:
6130:
6127:Totient rule
6120:
6116:
6112:
6108:
6107:consists of
6104:
6102:
6099:
5986:
5857:
5854:
5788:
5774:
5770:
5766:
5758:
5746:
5733:
5420:
5391:
5383:
5361:
5359:
5348:
5317:
5309:
5305:
5302:proper prime
5301:
5299:
5295:
5290:
5252:
5225:
5068:
5066:
4893:
4690:
4674:
4670:
4669:is equal to
4666:
4658:
4646:
4642:digital root
4640:The base-10
4639:
4632:
4631:10 ≡ 1 (mod
4622:
4618:
4614:
4606:
4594:
4570:
4538:
4532:
4528:
4522:
4514:
4506:
4498:
4496:
4475:
4471:
4463:
4449:
4447:
4426:
4422:
4414:
4402:
4381:
4373:
4361:
4355:
4343:
4335:
4317:
4310:
4308:
4287:
4279:
4265:
4258:
4257:The lengths
4256:
4248:
4240:
4223:
4221:
3765:
3753:
3321:
3309:
3188:000100111011
2766:
2751:
2739:
2603:
2527:
2421:
2254:
2249:
2245:
2244:denotes the
2216:
2152:
2081:digits, let
2000:
1942:
1743:
1740:Formal proof
1731:
1688:
1677:
1666:
1662:
1648:
1641:
1631:
1627:
1621:
1617:
1613:
1603:
1599:
1595:
1587:
1579:
1573:
1559:
1554:
1550:
1546:
1539:
1511:
1493:
1477:
1451:
1397:
1372:South Africa
1327:
1273:
555:
540:
531:
516:Please help
504:
443:
424:
420:
419:). However,
387:
382:of the form
359:of 10 (e.g.
348:
344:
340:
338:
317:
275:
270:
254:
250:
248:
233:
215:
206:
199:Please help
194:
183:
153:
144:
134:
127:
120:
113:
101:
89:Please help
84:verification
81:
57:
50:
44:
43:Please help
40:
15246:Cryptologia
15171:(499): 86.
14851:this length
14841:For a base
13671:which is 0.
12463:terminating
11318:Similarly,
10955:The symbol
7830:0.836363636
6706:The period
6649:The period
6626:(7 × 17) =
6557:is usually
6450:is usually
6259:are called
5360:If a prime
5304:is a prime
4571:A fraction
1687:Therefore,
1418:Netherlands
1398:Parentheses
1384:South Korea
1364:New Zealand
1302:Switzerland
1238:88679245283
1166:3.(142857)
1091:0.012345679
996:0.(142857)
577:Parentheses
271:terminating
203:if you can.
15360:Categories
15147:2023-12-23
15117:2023-12-23
15088:2023-12-23
14828:2023-12-19
13742:(and base
13146:= 0.3 and
13096:duodecimal
12658: := d
12033:For prime
10069:Examples:
9655:1234 − 123
8376:0.18181818
8357:7.48181818
8129:A shortcut
7895:8.36363636
6362:such that
5734:(sequence
5370:safe prime
4665:for prime
4501:for which
4452:for which
3761:expansion
3317:expansion
3114:0001011101
2762:expansion
2747:expansion
1390:, and the
852:0.8181...
481:Background
454:irrational
117:newspapers
46:improve it
15347:MathWorld
15322:1312.3618
15193:125834304
15014:∣
15001:
14970:λ
14920:≡
14906:∣
14897:∈
14873:
14524:−
14515:≤
14507:⟹
14496:≤
14452:−
14430:⟹
14413:−
14361:≤
14317:−
14209:substring
14176:substring
14077:substring
14014:substring
13646:⋯
13460:−
13451:−
13426:⋯
13402:−
13388:−
13373:⋯
13341:−
13300:−
13259:−
13063:−
13058:ℓ
13036:ℓ
13028:…
12963:¯
12957:ℓ
12949:…
12691:over the
12683:from the
12607:…
12522:−
12516:∖
12508:∈
12470:dense set
12321:⋅
12258:then for
12203:≠
12159:for some
12110:⋯
11556:−
11514:−
11481:−
11465:⟹
11435:∞
11420:∑
11413:⋯
11341:¯
11284:−
11256:−
11230:−
11200:converges
11128:∞
11113:∑
11106:⋯
11059:¯
11005:#
10963:∅
10893:−
10859:#
10843:#
10783:¯
10767:…
10764:0.3789789
10710:−
10676:#
10660:#
10639:∅
10600:¯
10584:…
10544:−
10510:#
10494:#
10434:¯
10418:…
10378:−
10344:#
10328:#
10307:∅
10268:¯
10252:…
10212:−
10178:#
10162:#
10102:¯
10086:…
10018:#
9973:#
9917:#
9870:#
9061:⋯
9043:⋯
9008:−
8983:⋯
8933:⋯
8897:⋯
8877:−
8853:¯
8839:⋯
8800:⋯
8751:¯
8737:⋯
8635:⋅
8597:…
8578:…
8522:⋅
8511:⋅
8499:⋅
8448:⋅
8437:⋅
8379:…
8360:…
8284:−
8221:−
8199:…
8169:…
8087:⋅
8076:⋅
7954:…
7898:…
7833:…
7634:…
7581:…
7394:⋯
7389:ℓ
7317:⋯
7312:ℓ
7278:−
7227:⋯
7222:ℓ
7188:−
7134:⋯
7129:ℓ
7101:⋅
6972:…
6942:ℓ
6906:
6874:⋯
6859:ℓ
6746:), where
6383:≡
6293:repeats:
6169:), where
5840:909090909
5825:−
5817:−
5355:congruent
4320:) of the
2686:−
2635:−
2581:−
2544:−
2504:−
2496:−
2447:∞
2432:∑
2402:−
2360:∞
2345:∑
2290:∞
2275:∑
2133:∈
2107:¯
2064:∈
2038:≠
1981:¯
1881:⌉
1874:
1860:⌈
1772:¯
1464:Argentina
1458:and some
1388:Singapore
1382:, India,
1368:Australia
1276:: In the
1093:012345679
1086:012345679
1053:012345679
1044:999999999
774:0.666...
706:0.333...
638:0.111...
565:Fraction
505:does not
209:July 2024
147:July 2023
52:talk page
15266:, Vol. I
14955:≡ 1 mod
14853:divides
14807:(4): 7–9
14761:0.999...
14725:See also
14540:′
14439:′
14355:⌋
14337:⌊
14291:we have
14260:function
14203:overline
13760:function
12851:divides
12693:alphabet
11855:, where
10249:0.512512
10083:3.254444
9690:3789 − 3
7631:3.333333
7578:0.333333
7347:, or as
7257:, or as
6620:= 7 × 17
6522:because
6459:, where
6421:that is
6161:divides
6115:, where
5662:01369863
5364:is both
5215:= 6 × 0.
5190:= 5 × 0.
5165:= 4 × 0.
5140:= 3 × 0.
5115:= 2 × 0.
5090:= 1 × 0.
4224:fraction
4222:Thereby
3793:0.03125
3754:fraction
3310:fraction
2740:fraction
1682:= 58086
1478:Ellipsis
1404:, incl.
1380:Thailand
1340:Pakistan
1332:Malaysia
1314:Slovenia
1310:Slovakia
1274:Vinculum
1176:3.142857
1038:12345679
1006:0.142857
911:0.58(3)
585:Ellipsis
569:Vinculum
534:May 2022
486:Notation
450:integers
402:1.585 =
361:1.585 =
341:repetend
278:rational
267:periodic
196:article.
184:require
15185:3621484
14715:
14700:
14620:
14606:
13844:defined
13738:
13722:
13720:0 <
13655:base 12
13196:
13184:
13176:
13164:
13160:
13148:
13144:
13132:
13130:= 0.4,
13128:
13116:
13114:= 0.6,
13112:
13100:
12828:
12816:
12687:of the
12487:If the
12428:
12413:
12411:=
12389:
12375:
12028:
12014:
12010:
11996:
11992:
11978:
11963:
11949:
11945:
11931:
11927:
11913:
11909:
11888:
11859:is the
11847:,
11843:is max(
11841:
11827:
11804:
11790:
11782:coprime
11775:
11759:
11748:
11734:
11719:
11705:
11686:
11672:
11656:
11644:
11640:
11628:
11196:
11184:
11180:
11168:
10415:1.09191
10041:integer
10035:times.
9808:
9796:
9780:
9768:
9753:
9737:
9715:
9703:
9699:
9687:
9680:
9668:
9664:
9652:
9639:
9627:
9623:
9611:
9607:
9595:
9591:
9579:
9575:
9563:
9557:
9545:
9541:
9529:
9525:
9513:
9509:
9497:
9493:
9481:
9477:
9465:
9457:
9445:
9441:
9429:
9425:
9413:
9409:
9397:
9393:
9381:
9375:
9363:
9359:
9347:
9343:
9331:
9327:
9315:
9311:
9299:
9286:
9274:
9270:
9258:
9251:
9239:
9232:
9220:
9192:
9180:
9173:
9161:
9157:
9145:
9138:
9126:
9119:
9107:
8336:
8324:
8301:9999999
8238:9999999
7528:
7516:
7482:
7470:
7462:
7444:
7435:,
7063:
7049:
7045:
7031:
7027:
7013:
6798:
6784:
6771:
6757:
6737:,
6728:is LCM(
6726:
6712:
6696:
6684:
6669:
6655:
6611:
6599:
6595:
6581:
6555:
6541:
6532:in the
6529:A045616
6520:
6506:
6502:
6488:
6484:
6470:
6448:
6434:
6423:coprime
6312:
6298:
6291:
6277:
6241:
6225:
6173:is the
6159:
6145:
6094:
6082:
6076:
6064:
6058:
6046:
6040:
6028:
6022:
6010:
6004:
5992:
5981:
5969:
5963:
5951:
5945:
5933:
5927:
5915:
5909:
5897:
5891:
5879:
5872:
5860:
5793:
5779:
5763:
5749:
5740:in the
5737:A006559
5724:
5712:
5702:
5690:
5680:
5668:
5658:
5646:
5636:
5624:
5614:
5602:
5592:
5580:
5570:
5558:
5548:
5536:
5526:
5514:
5504:
5492:
5482:
5470:
5460:
5448:
5438:
5426:
5408:in the
5405:A000353
5388:
5374:
5372:, then
5341:in the
5338:A073761
5329:
5314:
5287:
5275:
5271:
5259:
5248:142,857
5240:
5228:
5213:
5201:
5188:
5176:
5163:
5151:
5138:
5126:
5113:
5101:
5088:
5076:
5060:in the
5057:A001913
5052:
5040:
5036:
5024:
5020:
5008:
5004:
4992:
4988:
4976:
4972:
4960:
4956:
4944:
4940:
4928:
4924:
4912:
4908:
4896:
4884:
4872:
4862:
4850:
4840:
4828:
4818:
4806:
4796:
4784:
4774:
4762:
4752:
4740:
4730:
4718:
4708:
4696:
4663:
4649:
4613:modulo
4599:
4585:
4581:coprime
4575:with a
4552:in the
4549:A054471
4519:
4503:
4489:in the
4486:A007138
4468:
4454:
4440:in the
4437:A002371
4419:
4405:
4395:in the
4392:A036275
4378:
4364:
4352:A007733
4340:
4326:
4301:in the
4298:A051626
4284:
4270:
4245:
4231:
4226:is the
4203:
4191:
4175:
4163:
4147:
4135:
4119:
4107:
4091:
4079:
4063:
4051:
4035:
4023:
4010:
3998:
3982:
3970:
3954:
3942:
3926:
3914:
3898:
3886:
3870:
3858:
3842:
3830:
3814:
3802:
3789:
3777:
3759:decimal
3731:
3719:
3703:
3691:
3675:
3663:
3647:
3635:
3619:
3607:
3591:
3579:
3566:
3554:
3538:
3526:
3510:
3498:
3482:
3470:
3454:
3442:
3429:
3417:
3401:
3389:
3373:
3361:
3345:
3333:
3315:decimal
3294:0.0001
3288:0.0625
3284:
3272:
3247:
3235:
3210:
3198:
3173:
3161:
3136:
3124:
3099:
3087:
3065:
3053:
3028:
3016:
2997:
2985:
2960:
2948:
2923:
2911:
2889:
2877:
2858:
2846:
2821:
2809:
2790:
2778:
2745:decimal
2252:, and
1723:
1711:
1707:
1695:
1592:
1576:
1534:
1522:
1438:Vietnam
1434:Ukraine
1414:Finland
1410:Denmark
1406:Austria
1352:Algeria
1344:Tunisia
1336:Morocco
1294:Germany
1217:
1205:
1199:
1187:
1132:
1123:3142854
1120:
1114:
1102:
1068:1234567
1047:
1035:
1029:
1017:
962:
950:
944:
932:
888:
876:
870:
858:
843:0.(81)
810:
798:
792:
780:
742:
730:
724:
712:
674:
662:
656:
644:
604:
592:
526:removed
511:sources
464:√
448:of two
416:
404:
398:
384:
375:
363:
345:reptend
334:
322:
314:
302:
294:
282:
186:cleanup
131:scholar
15191:
15183:
14950:(with
14577:occurs
14278:modulo
14242:return
14158:occurs
14137:occurs
14083:digits
14044:return
14020:digits
13859:occurs
13850:occurs
13790:digits
13763:b_adic
13740:< 1
13708:base10
13706:is 125
13704:base12
13700:base10
13696:base12
13692:base10
13690:is 144
13688:base12
13684:base10
13680:base12
13676:base12
13673:186A35
13200:186A35
12891:λ
12648:|
12456:digits
12430:times.
12325:period
12273:period
12207:period
12174:period
12163:, but
12117:period
12078:period
12052:period
11817:where
11777:, for
11573:999999
11570:142857
11542:142857
11496:142857
11442:142857
11395:142857
11375:142857
11355:142857
11338:142857
9728:, and
9723:period
7892:
7889:
7886:
7883:
7827:
7824:
7821:
7818:
7486:571428
7484:= 0.03
7156:where
6990:where
5782:10 − 1
5486:076923
5221:857142
5217:142857
5196:714285
5192:142857
5171:571428
5167:142857
5146:428571
5142:142857
5121:285714
5117:142857
5096:142857
5092:142857
5069:proper
5067:Every
4712:142857
4322:binary
4069:238095
4014:0.025
3988:025641
3876:285714
3653:571428
3597:384615
3460:047619
3216:714285
3179:076923
3043:000111
3007:0.001
3001:0.125
2966:142857
2760:binary
2604:Since
2422:Since
2217:where
2122:where
1826:, and
1786:where
1563:0.0675
1472:Mexico
1470:, and
1468:Brazil
1442:Israel
1430:Russia
1426:Poland
1422:Norway
1416:, the
1402:Europe
1322:Turkey
1320:, and
1304:, the
1290:France
1282:Canada
1178:142857
1171:142857
1138:142857
1129:999999
1008:142857
1001:142857
968:142857
959:999999
953:142857
765:0.(6)
697:0.(3)
629:0.(1)
400:(e.g.
318:second
263:digits
133:
126:
119:
112:
104:
15318:arXiv
15189:S2CID
15181:JSTOR
14773:Notes
14570:while
14396:thus
14289:floor
14122:while
13899:floor
13838:while
13805:begin
13752:>1
13698:is 25
13682:is 12
12765:with
12639:with
12448:radix
12369:is a
11965:,....
10581:1.333
9267:99900
7507:⋯ = 7
7495:= 2,
7343:>
7253:>
6634:(7),
5552:02439
4603:order
4577:prime
4354:, if
4041:02439
3570:0.04
3542:0.041
3433:0.05
2868:0.01
2862:0.25
2250:digit
2053:with
1734:below
1698:58086
1661:10000
1598:, is
1571:....
1456:Spain
1454:: In
1376:Japan
1356:Egypt
1318:Chile
1298:Italy
1286:India
573:Dots
446:ratio
421:every
380:ratio
357:power
257:is a
138:JSTOR
124:books
14479:and
14461:<
14419:<
14325:<
14155:from
13999:then
13969:then
13702:, A5
13694:, 21
13686:, 10
13678:. 10
13628:1985
13198:= 0.
13180:2497
13178:= 0.
12880:mod
12778:and
12748:...(
12733:and
12670:0 ∈
12668:and
12478:and
12399:= 10
12037:, if
12012:and
11988:k k′
11972:and
11098:1000
10934:1665
10917:9990
10914:3786
10900:9990
10890:3789
10565:1081
10541:1091
10233:2929
10209:3254
10094:3.25
9782:has
9712:9990
9706:3786
9696:9990
9671:1111
9636:1665
9620:9990
9614:3786
9604:9990
9598:9780
9588:9990
9582:5994
9554:1665
9538:9990
9532:3786
9522:9990
9506:9990
9500:2997
9490:9990
9448:1111
9366:1111
9334:1107
9283:8325
9248:9900
9229:9000
7924:1000
7530:has.
7160:and
6773:and
6698:= 0.
6575:and
6534:OEIS
6314:= 0.
6243:for
6207:) =
5742:OEIS
5726:= 0.
5704:= 0.
5682:= 0.
5660:= 0.
5638:= 0.
5616:= 0.
5594:= 0.
5572:= 0.
5550:= 0.
5528:= 0.
5506:= 0.
5484:= 0.
5462:= 0.
5440:= 0.
5410:OEIS
5368:and
5353:and
5343:OEIS
5219:= 0.
5194:= 0.
5169:= 0.
5144:= 0.
5119:= 0.
5094:= 0.
5062:OEIS
4886:= 0.
4864:= 0.
4842:= 0.
4820:= 0.
4798:= 0.
4776:= 0.
4754:= 0.
4732:= 0.
4710:= 0.
4554:OEIS
4527:1 ≤
4521:has
4491:OEIS
4442:OEIS
4397:OEIS
4303:OEIS
4247:and
4123:0.02
3902:0.02
3651:0.03
3262:0001
3149:0.00
3140:0.08
3077:0011
3069:0.1
2901:0011
2893:0.2
2800:0.1
2794:0.5
1714:3227
1704:9990
1676:9990
1665:− 10
1630:and
1557:500
1547:4.44
1440:and
1432:and
1354:and
1348:Iran
1328:Dots
1266:...
1181:...
1153:4285
1096:...
1011:...
983:4285
926:...
921:0.58
914:0.58
898:0.58
892:0.58
581:Arc
509:any
507:cite
471:and
433:and
372:1000
366:1585
305:3227
110:news
15173:doi
15018:gcd
14992:ord
14985:max
14927:mod
14888:min
14864:ord
14673:mod
14663:mod
14595:).
14257:end
14239:for
14236:end
14164:pos
14149:for
14131:pos
14119:end
14107:pos
14059:end
14038:end
13987:not
13865:pos
13841:not
13823:pos
13608:441
13205:If
12863:ord
12856:− 1
12755:+1)
12715:...
12666:− 1
12472:in
12365:If
11968:If
11874:If
11809:If
11784:to
11728:If
11699:If
11085:100
10931:631
10835:789
10780:789
10775:0.3
10568:990
10551:990
10426:1.0
10402:999
10399:512
10385:999
10375:512
10320:512
10265:512
10236:900
10219:900
10215:325
9677:900
9661:900
9630:631
9548:631
9516:789
9484:789
9454:900
9438:900
9432:400
9422:900
9416:711
9390:100
9372:900
9356:900
9340:900
9324:900
9308:100
9302:123
9170:333
9154:999
8658:593
8560:or
8539:110
8536:823
8370:7.3
8062:990
8059:828
8007:828
7980:990
7416:if
7339:if
7249:if
6903:LCM
6803:If
6710:of
6693:119
6653:of
6628:LCM
6618:119
6608:119
6571:If
6536:).
6425:to
6394:mod
6271:If
6211:− 1
5530:027
5291:its
5064:).
5049:229
5033:223
5017:193
5001:181
4985:179
4969:167
4953:149
4937:131
4921:113
4905:109
4537:),
4213:46
4185:22
4179:0.0
4151:0.0
4101:21
4067:0.0
3964:18
3958:0.0
3932:027
3874:0.0
3852:16
3846:0.0
3741:15
3707:0.0
3685:28
3625:037
3595:0.0
3520:22
3486:0.0
3411:18
3377:0.0
3355:16
3251:0.0
3225:001
3223:0.0
3214:0.0
3192:12
3118:10
3075:0.0
2975:001
2936:0.0
2927:0.1
2704:),
2248:th
1865:log
1720:555
1606:).
1568:675
1565:675
1555:370
1551:518
1542:675
1540:0.0
1499:11.
1452:Arc
1254:11.
1227:11.
1221:11.
1190:593
923:333
885:900
879:525
520:by
441:.)
413:2·5
407:317
395:2·5
343:or
325:593
311:555
253:or
93:by
15362::
15344:.
15261:,
15195:.
15187:.
15179:.
15169:84
15167:.
15135:.
15105:.
15081:.
15067:).
14982::=
14885::=
14821:.
14805:60
14803:,
14721:.
14533:=:
14443::=
14274:p′
14269:.
14230:))
14173:do
14161:to
14110:+=
14062:if
14041:if
13984:if
13957:if
13905:bp
13874:bp
13856:do
13747:∈
13710:.
13632:10
13612:10
13592:10
13567:10
13563:21
13547:10
13523:10
13098:,
12923:0.
12786:≠
12773:∈
12735:0.
12702:0.
12662:+
12650:,
12569::=
12425:10
12024:k′
11974:k′
11959:p″
11947:,
11941:p′
11929:,
11904:p″
11901:p′
11884:p″
11882:,
11880:p′
11878:,
11658:.
11653:39
11647:16
11637:39
11547:10
11523:10
11500:10
11446:10
11403:18
11399:10
11383:12
11379:10
11359:10
11333:0.
11281:10
11264:10
11248:10
11193:10
11177:10
11139:10
11072:10
11051:0.
10731:12
10707:13
10592:1.
10547:10
10486:91
10431:91
10260:0.
10141:25
9763:.
9701:=
9666:=
9625:=
9609:=
9593:+
9572:10
9543:=
9527:=
9511:+
9495:=
9479:+
9474:10
9443:=
9427:+
9411:=
9395:+
9384:79
9361:=
9345:+
9329:=
9313:+
9272:=
9261:12
9242:56
9148:12
9135:99
9129:56
9064:99
9058:99
8999:10
8900:99
8894:99
8868:10
8761:10
8711:0.
8661:53
8648:53
8644:10
8638:53
8632:11
8621:53
8618:10
8610:11
8588:11
8525:11
8519:10
8508:10
8502:73
8496:11
8478:11
8465:10
8462:73
8451:11
8423:10
8420:73
8410:99
8407:18
8397:10
8394:73
8275:10
8212:10
8177:10
8104:55
8101:46
8090:55
8084:18
8079:46
8073:18
7856:10
7604:10
7488::
7479:28
7424:.
7420:=
7365:10
7288:10
7198:10
7047:,
7029:,
7004:,
6997:,
6823:,
6819:,
6811:,
6807:,
6800:.
6722:pq
6677:pq
6665:pq
6646:.
6613::
6591:pq
6452:pT
6429:.
6308:49
6263:.
6195:.
6117:nk
6091:13
6073:13
6055:13
6049:11
6037:13
6019:13
6001:13
5978:13
5960:13
5942:13
5936:12
5924:13
5906:13
5900:10
5888:13
5869:13
5832:11
5814:11
5810:10
5744:)
5721:89
5699:83
5677:79
5655:73
5633:71
5611:67
5589:53
5567:43
5545:41
5523:37
5501:31
5479:13
5464:09
5457:11
5412:).
5345:).
5326:10
5300:A
5038:,
5022:,
5006:,
4990:,
4974:,
4958:,
4942:,
4926:,
4910:,
4881:97
4859:61
4837:59
4815:47
4793:29
4771:23
4749:19
4727:17
4677:.
4637:.
4556:).
4535:−1
4531:≤
4493:).
4474:,
4444:).
4421:,
4399:).
4380:,
4342:,
4305:).
4286:,
4262:10
4252:10
4207:0.
4200:47
4172:46
4157:1
4144:45
4129:2
4125:27
4116:44
4095:0.
4088:43
4073:6
4060:42
4045:5
4039:0.
4032:41
4017:0
4007:40
3992:6
3986:0.
3979:39
3951:38
3936:3
3930:0.
3923:37
3908:1
3895:36
3880:6
3867:35
3839:34
3824:2
3820:03
3818:0.
3811:33
3796:0
3786:32
3769:10
3735:0.
3728:31
3713:1
3700:30
3679:0.
3672:29
3657:6
3644:28
3629:3
3623:0.
3616:27
3601:6
3588:26
3573:0
3563:25
3548:1
3535:24
3514:0.
3507:23
3492:2
3488:45
3479:22
3464:6
3458:0.
3451:21
3436:0
3426:20
3405:0.
3398:19
3383:1
3370:18
3349:0.
3342:17
3325:10
3297:0
3291:0
3281:16
3266:4
3260:0.
3257:1
3244:15
3229:3
3220:6
3207:14
3186:0.
3183:6
3177:0.
3170:13
3155:2
3151:01
3146:1
3133:12
3112:0.
3109:2
3105:09
3103:0.
3096:11
3081:4
3072:0
3062:10
3047:6
3041:0.
3038:1
3032:0.
3010:0
3004:0
2979:3
2973:0.
2970:6
2964:0.
2942:2
2938:01
2933:1
2905:4
2899:0.
2896:0
2871:0
2865:0
2840:2
2836:01
2834:0.
2831:1
2825:0.
2803:0
2797:0
2755:10
2677:10
2662:10
2572:10
2557:10
2525:,
2500:10
2463:10
2376:10
2306:10
2246:i-
1951:10
1922:10
1869:10
1806:,
1736:.
1709:=
1693:=
1536::
1531:74
1496:34
1484:;
1466:,
1428:,
1424:,
1420:,
1412:,
1408:,
1386:,
1378:,
1374:,
1370:,
1366:,
1362:,
1350:,
1346:,
1342:,
1338:,
1334:,
1316:,
1312:,
1308:,
1300:,
1296:,
1292:,
1288:,
1284:,
1280:,
1203:=
1196:53
1169:3.
1142:3.
1136:3.
1118:=
1105:22
1084:0.
1057:0.
1051:0.
1033:=
1026:81
999:0.
972:0.
966:0.
948:=
874:=
867:12
848:81
846:0.
820:0.
816:81
814:0.
807:99
801:81
796:=
789:11
768:0.
752:0.
746:0.
728:=
700:0.
684:0.
678:0.
660:=
632:0.
616:0.
610:0.
477:.
331:53
249:A
55:.
15350:.
15320::
15175::
15150:.
15120:.
15091:.
15065:n
15063:(
15061:φ
15042:}
15039:1
15036:=
15033:)
15030:n
15027:,
15024:b
15021:(
15010:)
15007:b
15004:(
14996:n
14988:{
14979:)
14976:n
14973:(
14957:n
14936:}
14931:n
14923:1
14915:L
14911:b
14901:N
14894:L
14891:{
14882:)
14879:b
14876:(
14868:n
14847:n
14843:b
14831:.
14712:2
14709:/
14704:p
14696:p
14677:2
14667:p
14657:i
14653:2
14649:=
14646:)
14643:i
14640:(
14637:a
14624:p
14616:p
14612:/
14609:1
14589:L
14585:p
14581:z
14566:q
14562:p
14545:.
14537:p
14530:q
14527:z
14521:p
14518:b
14512:0
14502:p
14499:b
14493:q
14490:z
14464:q
14458:q
14455:z
14449:p
14446:b
14436:p
14425:q
14422:z
14416:q
14410:p
14407:b
14381:,
14376:q
14372:p
14369:b
14350:q
14346:p
14343:b
14333:=
14330:z
14320:1
14312:q
14308:p
14305:b
14282:q
14267:z
14254:;
14251:)
14248:s
14245:(
14233:;
14227:1
14224:,
14221:i
14218:,
14215:s
14212:(
14206:(
14200:=
14197:)
14194:1
14191:,
14188:i
14185:,
14182:s
14179:(
14170:1
14167:-
14152:i
14140:;
14134:-
14128:=
14125:L
14116:;
14113:1
14101:;
14098:)
14095:1
14092:,
14089:z
14086:,
14080:(
14074:.
14071:s
14068:=
14065:s
14056:;
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14050:s
14047:(
14035:)
14032:1
14029:,
14026:z
14023:,
14017:(
14011:.
14008:s
14005:=
14002:s
13996:0
13993:=
13990:z
13981:;
13978:0
13975:=
13972:L
13966:0
13963:=
13960:p
13950:;
13947:q
13944:*
13941:z
13939:−
13937:p
13934:*
13931:b
13928:=
13925:p
13917:;
13914:)
13911:q
13908:/
13902:(
13896:=
13893:z
13889:;
13886:p
13883:*
13880:b
13877:=
13868:;
13862:=
13853:)
13847:(
13832:;
13829:0
13826:=
13817:;
13811:=
13808:s
13799:;
13793:=
13784:)
13781:q
13778:,
13775:p
13772:,
13769:b
13766:(
13749:N
13745:b
13734:q
13730:/
13726:p
13650:)
13643:+
13636:6
13623:+
13616:5
13603:+
13596:4
13587:5
13584:A
13578:+
13571:3
13558:+
13551:2
13543:5
13538:+
13529:1
13519:1
13513:(
13508:=
13503:7
13500:1
13475:.
13467:b
13463:k
13457:b
13448:1
13444:1
13437:b
13434:1
13429:=
13423:+
13416:N
13412:b
13405:1
13399:N
13395:)
13391:k
13385:b
13382:(
13376:+
13370:+
13363:4
13359:b
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13348:)
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13329:+
13322:3
13318:b
13311:2
13307:)
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13288:+
13281:2
13277:b
13270:1
13266:)
13262:k
13256:b
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13247:+
13242:b
13239:1
13234:=
13229:k
13226:1
13211:k
13207:b
13193:7
13190:/
13187:1
13173:5
13170:/
13167:1
13157:6
13154:/
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13122:/
13119:1
13109:2
13106:/
13103:1
13072:.
13066:1
13054:b
13046:b
13042:)
13032:A
13023:2
13019:A
13013:1
13009:A
13005:(
12974:b
12969:)
12953:A
12944:2
12940:A
12934:1
12930:A
12919:(
12901:q
12897:)
12895:q
12893:(
12882:q
12878:b
12873:)
12871:b
12869:(
12866:q
12854:b
12849:q
12845:l
12841:q
12837:l
12830:.
12825:2
12822:/
12819:1
12812:T
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