887:
1705:
8501:
1496:
2328:
7450:
A function with a vertical asymptote is not helpful in defining a very large number, although the function increases very rapidly: one has to define an argument very close to the asymptote, i.e. use a very small number, and constructing that is equivalent to constructing a very large number, e.g. the
5587:
3507:
If the superscript of the upward arrow is large, the various representations for large numbers can be applied to this superscript itself. If this superscript is not exactly given then there is no point in raising the operator to a particular power or to adjust the value on which it act, instead it is
7913:
4811:
For a number too large to write down in the Conway chained arrow notation it size can be described by the length of that chain, for example only using elements 10 in the chain; in other words, one could specify its position in the sequence 10, 10β10, 10β10β10, .. If even the position in the sequence
2025:
of the whole number. If the power tower would contain one or more numbers different from 10, the two approaches would lead to different results, corresponding to the fact that extending the power tower with a 10 at the bottom is then not the same as extending it with a 10 at the top (but, of course,
109:
10, signifies one billionth, equivalent to 0.000β000β001. By using 10 instead of explicitly writing out all those zeros, readers are spared the effort and potential confusion of counting an extended series of zeros to grasp the magnitude of the number. Additionally, alongside scientific notation
11527:
8070:
9813:
One solution to the problem of comparing large numbers is to define classes of numbers, such as the system devised by Robert Munafo, which is based on different "levels" of perception of an average person. Class 0 β numbers between zero and six β is defined to contain numbers that are easily
8331:
3494:
9050:
7087:
9145:
8877:
9818:, that is, numbers that show up very frequently in daily life and are almost instantly comparable. Class 1 β numbers between six and 1,000,000=10 β is defined to contain numbers whose decimal expressions are easily subitized, that is, numbers who are easily comparable not by
8601:
8953:
8337:
5073:
5334:
5785:
6540:
9825:
Each class after these are defined in terms of iterating this base-10 exponentiation, to simulate the effect of another "iteration" of human indistinguishibility. For example, class 5 is defined to include numbers between 10 and 10, which are numbers where
5201:
1700:{\displaystyle 2\uparrow \uparrow \uparrow 4={\begin{matrix}\underbrace {2_{}^{2^{{}^{.\,^{.\,^{.\,^{2}}}}}}} \\\qquad \quad \ \ \ 65,536{\mbox{ copies of }}2\end{matrix}}\approx (10\uparrow )^{65,531}(6\times 10^{19,728})\approx (10\uparrow )^{65,533}4.3}
2742:
11542:
8223:
4689:): it is the general process of adding an element 10 to the chain in the chain notation; this process can be repeated again (see also the previous section). Numbering the subsequent versions of this function a number can be described using functions
8729:
950:
To compare numbers in scientific notation, say 5Γ10 and 2Γ10, compare the exponents first, in this case 5 > 4, so 2Γ10 > 5Γ10. If the exponents are equal, the mantissa (or coefficient) should be compared, thus 5Γ10 > 2Γ10 because 5 > 2.
2029:
If the height of the tower is large, the various representations for large numbers can be applied to the height itself. If the height is given only approximately, giving a value at the top does not make sense, so the double-arrow notation (e.g.
7446:
These notations are essentially functions of integer variables, which increase very rapidly with those integers. Ever-faster-increasing functions can easily be constructed recursively by applying these functions with large integers as argument.
8799:
4039:
2583:
is large, the various representations for large numbers can be applied to this exponent itself. If this exponent is not exactly given then, again, giving a value at the right does not make sense, and instead of using the power notation of
270:(approximately 8.8βΓβ10^26 meters) and hosts around 5βΓβ10^22 stars, organized into roughly 125 billion galaxies (as observed by the Hubble Space Telescope). As a rough estimate, there are about 10^80 atoms within the observable universe.
7022:
6966:
6910:
6854:
6801:
6748:
6607:
5885:
6696:
2200:
1363:
5410:
3702:
If the number of levels gets too large to be convenient, a notation is used where this number of levels is written down as a number (like using the superscript of the arrow instead of writing many arrows). Introducing a function
7748:
3414:
6211:
6080:
1876:. As explained, a more precise description of a number also specifies the value of this number between 1 and 10, or the previous number (taking the logarithm one time less) between 10 and 10, or the next, between 0 and 1.
9638:
to be "close in magnitude". The relative error between these two numbers is large, and the relative error between their logarithms is still large; however, the relative error in their second-iterated logarithms is small:
7919:
3941:
2806:
2774:
9977:
8229:
6475:
6422:
6369:
6316:
6263:
6132:
5963:
5666:
10126:
2141:
7643:
3419:
8960:
7029:
1486:
1236:
1021:
9409:
9055:
2656:
1958:
9260:
too large or too small. This seems like extremely poor accuracy, but for such a large number it may be considered fair (a large error in a large number may be "relatively small" and therefore acceptable).
8806:
3697:
2549:
7742:
9414:
a large relative error. However, one can also consider the relative error in the logarithms; in this case, the logarithms (to base 10) are 10 and 9, so the relative error in the logarithms is only 10%.
8506:
346:
8496:{\displaystyle 100\uparrow \uparrow \uparrow n=(10\uparrow \uparrow )^{n-2}(10\uparrow )^{98}(2\times 10^{200})=(10\uparrow \uparrow )^{n-2}(10\uparrow )^{100}2.3<10\uparrow \uparrow \uparrow (n+1)}
8882:
9718:
7261:
3128:
2073:
11709:
6635:
9796:
9986:
A very large number raised to a very large power is "approximately" equal to the larger of the following two values: the first value and 10 to the power the second. For example, for very large
4949:
3563:
5404:
754:
8631:
7689:
5218:
8657:
2380:
2354:
2193:
2167:
7561:
6004:
5672:
4279:, etc. can be introduced. If many such functions are required, they can be numbered instead of using a new letter every time, e.g. as a subscript, such that there are numbers of the form
40:, far beyond those encountered in everyday lifeβsuch as simple counting or financial transactionsβplay a crucial role in various domains. These expansive quantities appear prominently in
11242:
6482:
4103:
3060:
1769:
1737:
10423:, representing the limit of all finite numbers k). This is the basis for the fast-growing hierarchy of functions, in which the indexing subscript is extended to ever-larger ordinals.
9231:
5088:
4802:
4520:
3186:
1874:
10386:
10328:
7169:
4419:
9564:
will have an even larger relative error. The question then becomes: on which level of iterated logarithms to compare two numbers? There is a sense in which one may want to consider
8100:
3749:
3014:
2611:
2581:
2491:
10248:
8126:
3846:
2965:
2664:
1836:
1288:
1262:
11371:
Bianconi, Eva; Piovesan, Allison; Facchin, Federica; Beraudi, Alina; Casadei, Raffaella; Frabetti, Flavia; Vitale, Lorenza; Pelleri, Maria Chiara; Tassani, Simone (NovβDec 2013).
947:
A standardized way of writing very large numbers allows them to be easily sorted in increasing order, and one can get a good idea of how much larger a number is than another one.
10088:
10044:
8133:
2004:
1156:
841:
8664:
4736:
4641:
3337:
1057:
800:
7499:
702:
11201:
4466:
2883:
3638:
4880:
4217:
3601:
2836:
2442:
2413:
9599:
8734:
4318:
1395:
9633:
9559:
9525:
4683:
4569:
1103:
3283:
635:
608:
10181:
4257:
4146:
3948:
3793:
1810:
575:
548:
516:
9301:
9258:
7208:
7129:
3248:
3219:
9485:
9458:
9328:
9185:
4604:
373:. This time assumes a statistical model subject to PoincarΓ© recurrence. A much simplified way of thinking about this time is in a model where the universe's history
10413:
9233:, with the 6.2 the result of proper rounding using significant figures, the true value of the exponent may be 50 less or 50 more. Hence the result may be a factor
6973:
6917:
6861:
6808:
6755:
6702:
6561:
2323:{\displaystyle 10\uparrow \uparrow 10\uparrow \uparrow (10\uparrow )^{497}(9.73\times 10^{32})=(10\uparrow \uparrow )^{2}(10\uparrow )^{497}(9.73\times 10^{32})}
10004:
5791:
2462:
5582:{\displaystyle 2^{2^{2^{2^{2^{2}}}}}=2\uparrow \uparrow 6=2^{2^{65,536}}\approx 2^{(10\uparrow )^{2}4.3}\approx 10^{(10\uparrow )^{2}4.3}=(10\uparrow )^{3}4.3}
7908:{\displaystyle 100\uparrow \uparrow 4=(10\uparrow )^{2}(2\times 10^{200}+0.3)=(10\uparrow )^{2}(2\times 10^{200})=(10\uparrow )^{3}200.3=(10\uparrow )^{4}2.3}
6640:
1293:
4160:, i.e., the number of levels of upward arrows is itself represented in the superscripted upward-arrow notation, etc. Using the functional power notation of
361:
for the quantum state of a hypothetical box containing a black hole with the estimated mass of the entire universe, observable or not, assuming a certain
3349:
8065:{\displaystyle 100\uparrow \uparrow n=(10\uparrow )^{n-2}(2\times 10^{200})=(10\uparrow )^{n-1}200.3=(10\uparrow )^{n}2.3<10\uparrow \uparrow (n+1)}
6139:
868:
is a large number named after AgustΓn Rayo which has been claimed to be the largest named number. It was originally defined in a "big number duel" at
6010:
4927:
googol = 10 = 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
8326:{\displaystyle 100\uparrow \uparrow \uparrow 3=10\uparrow \uparrow (10\uparrow )^{98}(2\times 10^{200})=10\uparrow \uparrow (10\uparrow )^{100}2.3}
3863:
11263:
2779:
2747:
9871:
6429:
6376:
6323:
6270:
6217:
6086:
5917:
5593:
3489:{\displaystyle 10\uparrow (10\uparrow \uparrow \uparrow 3)=10\uparrow \uparrow (10\uparrow \uparrow 10+1)\approx 10\uparrow \uparrow \uparrow 3}
11581:(1995), S. A. Fulling (ed), p. 461. Discourses in Mathematics and its Applications, No. 4, Texas A&M University Department of Mathematics.
10093:
9045:{\displaystyle 100\uparrow \uparrow \uparrow \uparrow n=(10\uparrow \uparrow \uparrow )^{n-2}(10\uparrow \uparrow )^{98}(10\uparrow )^{100}2.3}
7459:
The following illustrates the effect of a base different from 10, base 100. It also illustrates representations of numbers and the arithmetic.
7082:{\displaystyle 10\uparrow \uparrow \uparrow \uparrow \uparrow 2=10\uparrow \uparrow \uparrow \uparrow 10=(10\uparrow \uparrow \uparrow )^{10}1}
2084:
7568:
1407:
1161:
957:
9140:{\displaystyle 10\uparrow \uparrow \uparrow \uparrow n=(10\uparrow \uparrow \uparrow )^{n-2}(10\uparrow \uparrow )^{8}(10\uparrow )^{10}1}
1116:
in a generalized sense. A crude way of specifying how large a number is, is specifying between which two numbers in this sequence it is.
277:, physicist at the University of Alberta, Canada, the longest finite time that has so far been explicitly calculated by any physicist is
9339:
468:
9273:
may be large, yet there may still be a sense in which one wants to consider the numbers as "close in magnitude". For example, consider
8872:{\displaystyle 100\uparrow \uparrow \uparrow \uparrow 3=10\uparrow \uparrow \uparrow (10\uparrow \uparrow )^{98}(10\uparrow )^{100}2.3}
11620:
2075:) can be used. If the value after the double arrow is a very large number itself, the above can recursively be applied to that value.
12192:
11116:
3945:
Compare the definition of Graham's number: it uses numbers 3 instead of 10 and has 64 arrow levels and the number 4 at the top; thus
2616:
1885:
11348:
8596:{\displaystyle 10\uparrow \uparrow \uparrow n=(10\uparrow \uparrow )^{n-2}(10\uparrow )^{10}1<10\uparrow \uparrow \uparrow (n+1)}
3643:
227:
Note that this value of the
Shannon number is for Standard Chess. It has even larger values for larger-board chess variants such as
2496:
381:; this is the time scale when it will first be somewhat similar (for a reasonable choice of "similar") to its current state again.
8948:{\displaystyle 10\uparrow \uparrow \uparrow \uparrow 3=10\uparrow \uparrow \uparrow (10\uparrow \uparrow )^{8}(10\uparrow )^{10}1}
7695:
3569:. For such numbers the advantage of using the upward arrow notation no longer applies, so the chain notation can be used instead.
11812:
11098:
289:
1775:
Thus the "order of magnitude" of a number (on a larger scale than usually meant), can be characterized by the number of times (
9645:
7213:
3068:
2033:
869:
11656:
10129:
6614:
9723:
6552:= 3 ββββ 3 = 3 βββ (3 βββ 3) β 3 βββ (10 ββ 7.6 Γ 10) β 10 βββ (10 ββ 7.6 Γ 10) is between (10 βββ) 2 and (10 βββ) 3 (See
5068:{\displaystyle 2^{2^{2^{2^{2}}}}=2\uparrow \uparrow 5=2^{65,536}\approx 2.0\times 10^{19,728}\approx (10\uparrow )^{2}4.3}
7398:( 10 β 10 β 10 β 10 β 10 β 10 β 10 β ... β 10 β 10 β 10 β 10 β 10 β 10 β 10 β 10 ) where there are ( 10 β 10 β 10 ) "10"s
5329:{\displaystyle 3^{3^{3^{3}}}=3\uparrow \uparrow 4\approx 1.26\times 10^{3,638,334,640,024}\approx (10\uparrow )^{3}1.10}
4921:
12300:
12264:
3511:
5350:
2415:
is not exactly given then giving a value at the right does not make sense, and instead of using the power notation of
712:
12086:
11769:
11594:
11288:
8610:
7655:
5780:{\displaystyle 3^{3^{3^{3^{3}}}}=3\uparrow \uparrow 5\approx 3^{10^{3.6\times 10^{12}}}\approx (10\uparrow )^{4}1.10}
934:
8636:
6535:{\displaystyle 10\uparrow \uparrow \uparrow \uparrow 2=10\uparrow \uparrow \uparrow 10=(10\uparrow \uparrow )^{10}1}
2359:
2333:
2172:
2146:
916:
7506:
11906:
11901:
11896:
11891:
11886:
11881:
11876:
11871:
5970:
5196:{\displaystyle M_{82,589,933}\approx 1.49\times 10^{24,862,047}\approx 10^{10^{7.3955}}=(10\uparrow )^{2}\ 7.3955}
4924:= 10 = 100,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
402:
In statistical mechanics, combinatorial numbers reach such immense magnitudes that they are often expressed using
93:
was devised to manage the vast range of values encountered in scientific research. For instance, when we write 1.0
11210:
4044:
1106:
11491:
3852:
is large, any of the above can be used for expressing it. The "roundest" of these numbers are those of the form
1742:
1710:
12382:
11268:
11258:
11142:
9194:
4808:(where all these numbers are exactly given integers) with at the end a number in ordinary scientific notation.
4757:
4526:(where all these numbers are exactly given integers) with at the end a number in ordinary scientific notation.
4475:
3141:
1841:
912:
908:
413:
7437:. Alternative notations, like a more conventional function notation, can also be used with the same functions.
2737:{\displaystyle 10\uparrow \uparrow \uparrow (10\uparrow \uparrow )^{2}(10\uparrow )^{497}(9.73\times 10^{32})}
358:
12341:
12185:
12121:
12001:
10766:
10333:
10264:
7425:
7134:
4355:
2846:
432:
31:
8079:
3706:
2973:
2587:
2557:
2467:
12356:
12351:
12006:
11996:
11936:
11204:
10190:
8218:{\displaystyle 100\uparrow \uparrow \uparrow 2=(10\uparrow )^{98}(2\times 10^{200})=(10\uparrow )^{100}2.3}
8105:
7411:
3802:
2924:
1815:
1267:
1241:
17:
9862:
There are some general rules relating to the usual arithmetic operations performed on very large numbers:
8724:{\displaystyle 100\uparrow \uparrow \uparrow \uparrow 2=(10\uparrow \uparrow )^{98}(10\uparrow )^{100}2.3}
2026:
similar remarks apply if the whole power tower consists of copies of the same number, different from 10).
11951:
11805:
10053:
10009:
2661:
If the right-hand argument of the triple arrow operator is large the above applies to it, obtaining e.g.
11618:
Carl Sagan takes questions more from his 'Wonder and
Skepticism' CSICOP 1994 keynote, Skeptical Inquirer
10047:
1970:
1122:
805:
258:, we encounter staggering numbers related to length and time. For instance, according to the prevailing
12096:
7433:; apart from the method of construction of large numbers, this also involves a graphical notation with
4692:
4613:
3288:
1026:
764:
11711:, so starting the 64 steps with 1 instead of 4 more than compensates for replacing the numbers 3 by 10
11565:
7464:
662:
467:
quipped "billions and billions". The phrase has, however, now become a humorous fictitious numberβthe
12081:
11946:
11182:
8794:{\displaystyle 10\uparrow \uparrow \uparrow \uparrow 2=(10\uparrow \uparrow )^{8}(10\uparrow )^{10}1}
4428:
2855:
855:
445:
428:
3610:
12178:
12159:
11278:
9866:
The sum and the product of two very large numbers are both "approximately" equal to the larger one.
7430:
4844:
4263:
is given exactly and n is an integer which may or may not be given exactly. For example, if (10β10β
4171:
3579:
2814:
2418:
2389:
897:
10659:
9570:
4282:
1368:
1158:, i.e., with a power tower of 10s, and a number at the top, possibly in scientific notation, e.g.
262:, our universe is approximately 13.8 billion years old (equivalent to 4.355βΓβ10^17 seconds). The
110:
based on powers of 10, there exists systematic nomenclature for large numbers in the short scale
10510:
9604:
9530:
9496:
6553:
4650:
4536:
4034:{\displaystyle G<3\rightarrow 3\rightarrow 65\rightarrow 2<(10\to 10\to 65\to 2)=f^{65}(1)}
2808:). This can be done recursively, so it is possible to have a power of the triple arrow operator.
1066:
901:
427:
has made significant contributions to the study of very large numbers, including work related to
3138:
can also be very large, in general it can be written instead a number with a sequence of powers
12377:
12310:
12274:
12154:
12049:
11798:
11313:
10142:
9802:
7440:
3253:
1398:
613:
586:
374:
102:
85:
11359:
10150:
4226:
4115:
3762:
1782:
553:
526:
494:
12346:
12116:
12106:
12044:
9279:
9236:
7186:
7107:
7017:{\displaystyle 10\uparrow \uparrow \uparrow \uparrow 9=(10\uparrow \uparrow \uparrow )^{9}1=}
6961:{\displaystyle 10\uparrow \uparrow \uparrow \uparrow 8=(10\uparrow \uparrow \uparrow )^{8}1=}
6905:{\displaystyle 10\uparrow \uparrow \uparrow \uparrow 7=(10\uparrow \uparrow \uparrow )^{7}1=}
4739:
3224:
3195:
3027:
454:
396:
362:
53:
9463:
9436:
9306:
9163:
6849:{\displaystyle 10\uparrow \uparrow \uparrow \uparrow 6=(10\uparrow \uparrow \uparrow )^{6}1}
6796:{\displaystyle 10\uparrow \uparrow \uparrow \uparrow 5=(10\uparrow \uparrow \uparrow )^{5}1}
6743:{\displaystyle 10\uparrow \uparrow \uparrow \uparrow 4=(10\uparrow \uparrow \uparrow )^{4}1}
6602:{\displaystyle 10\uparrow \uparrow \uparrow \uparrow 3=(10\uparrow \uparrow \uparrow )^{3}1}
4582:
3508:
possible to simply use a standard value at the right, say 10, and the expression reduces to
11617:
11303:
11245:
10391:
9419:
5912:
3 βββ 3 = 3 ββ (3 ββ 3) β 3 ββ 7.6 Γ 10 β 10 ββ 7.6 Γ 10 is between (10 ββ) 2 and (10 ββ) 3
3603:
is obtained in the superscript of the first arrow, etc., or a nested chain notation, e.g.:
274:
10330:, each sequence growing much faster than the one before it. Thus it is possible to define
5880:{\displaystyle 2^{2^{2^{2^{2^{2^{2}}}}}}=2\uparrow \uparrow 7\approx (10\uparrow )^{4}4.3}
2917:, there is a natural notation for powers of this function (just like when writing out the
420:
numbers surpass even the GΓΆdel numbers associated with typical mathematical propositions.
8:
11991:
4425:
is large any of the above can be used to express it. Thus is obtained a nesting of forms
3799:
is given exactly and n is an integer which may or may not be given exactly (for example:
263:
214:
90:
81:
73:
6691:{\displaystyle \approx (10\uparrow \uparrow \uparrow )^{2}(10\uparrow \uparrow )^{3}154}
3342:
For describing numbers approximately, deviations from the decreasing order of values of
1358:{\displaystyle 10\uparrow \uparrow n<(10\uparrow )^{n}a<10\uparrow \uparrow (n+1)}
12336:
12215:
12039:
11941:
11931:
11582:
11464:
11437:
11418:
11273:
11156:
10632:
9989:
7419:
2447:
1113:
859:
845:
758:
378:
64:
delves into the naming conventions and properties of these immense numerical entities.
45:
11745:
11606:
11147:
Although all the numbers discussed above are very large, they are all still decidedly
9854:)), and finally an extremely long decimal expansion whose length can't be subitized).
5904:
2 ββββ 3 = 2 βββ 4 = 2 ββ 65,536 β (10 β) 4.3 is between 10 ββ 65,533 and 10 ββ 65,534
11956:
11721:
11590:
11499:
11469:
11410:
11402:
2811:
Then it is possible to proceed with operators with higher numbers of arrows, written
181:
151:
77:
11422:
416:, are vastβeven for mathematical statements of moderate length. Remarkably, certain
395:
of a fixed set of objects, grows exponentially as the number of objects increases.
12255:
12141:
11961:
11911:
11636:
11459:
11449:
11394:
11384:
10258:
4328:
are given exactly and n is an integer which may or may not be given exactly. Using
1060:
865:
476:
459:
3409:{\displaystyle 10\uparrow (10\uparrow \uparrow )^{5}a=(10\uparrow \uparrow )^{6}a}
409:
12320:
12101:
11624:
11454:
11389:
11372:
11128:
6206:{\displaystyle (10\uparrow \uparrow )^{2}(10\uparrow )^{497}(9.73\times 10^{32})}
424:
12230:
12225:
12201:
12131:
12022:
11921:
11523:
11293:
11172:
11138:
11134:
10420:
9270:
6075:{\displaystyle (10\uparrow \uparrow )^{2}10^{\,\!10^{10^{3.81\times 10^{17}}}}}
5204:
2842:
656:
259:
236:
221:
118:
4275:
is large any of the above can be used for expressing it. Similarly a function
12371:
12091:
11866:
11846:
11503:
11406:
11101:
function. Its value for even relatively small input is huge. The values of Ξ£(
7415:
4934:
464:
384:
228:
3936:{\displaystyle (10\to 10\to 3\to 2)=10\uparrow ^{10\uparrow ^{10^{10}}10}10}
3504:
and 10 are "almost equal" (for arithmetic of large numbers see also below).
852:). However, it can be represented using layers of Knuth's up-arrow notation.
12126:
11487:
11473:
11414:
11318:
11168:
10136:
4533:
is too large to be given exactly, the number concerned can be expressed as
2801:{\displaystyle 10\uparrow \uparrow \uparrow 10\uparrow \uparrow \uparrow 5}
2769:{\displaystyle 10\uparrow \uparrow \uparrow 10\uparrow \uparrow \uparrow 4}
641:
580:
417:
185:
154:
is the number of βelementary entitiesβ (usually atoms or molecules) in one
49:
9972:{\displaystyle (10^{a})^{\,\!10^{b}}=10^{a10^{b}}=10^{10^{b+\log _{10}a}}}
6470:{\displaystyle 10\uparrow \uparrow \uparrow 9=(10\uparrow \uparrow )^{9}1}
6417:{\displaystyle 10\uparrow \uparrow \uparrow 8=(10\uparrow \uparrow )^{8}1}
6364:{\displaystyle 10\uparrow \uparrow \uparrow 7=(10\uparrow \uparrow )^{7}1}
6311:{\displaystyle 10\uparrow \uparrow \uparrow 6=(10\uparrow \uparrow )^{6}1}
6258:{\displaystyle 10\uparrow \uparrow \uparrow 5=(10\uparrow \uparrow )^{5}1}
6127:{\displaystyle 10\uparrow \uparrow \uparrow 4=(10\uparrow \uparrow )^{4}1}
5958:{\displaystyle 10\uparrow \uparrow \uparrow 3=(10\uparrow \uparrow )^{3}1}
5661:{\displaystyle 10^{10^{10^{10}}}=10\uparrow \uparrow 4=(10\uparrow )^{4}1}
113:
Examples of large numbers describing everyday real-world objects include:
12305:
12269:
12136:
12076:
11323:
11308:
11283:
11164:
11094:
10121:{\displaystyle 2\uparrow \uparrow 65536\approx 10\uparrow \uparrow 65533}
9819:
7645:, the highest exponent is very little more than doubled (increased by log
3024:
and only in special cases the long nested chain notation is reduced; for
2136:{\displaystyle 10\uparrow \uparrow 10^{\,\!10^{10^{3.81\times 10^{17}}}}}
1105:(the function also expressed by the suffix "-plex" as in googolplex, see
392:
370:
220:
The lower bound on the game-tree complexity of chess, also known as the β
155:
41:
11398:
12284:
11926:
11586:
11577:
Information Loss in Black Holes and/or
Conscious Beings?, Don N. Page,
11160:
11148:
9815:
7638:{\displaystyle 100^{100^{100^{12}}}\approx 10^{10^{2*10^{24}+0.30103}}}
3221:) with at the end a number in ordinary scientific notation. Whenever a
706:
520:
450:
267:
4912:
10 = 1,000,000,000,000,000,000 = 1 billion billion = 1 quintilion
3496:. Thus is obtained the somewhat counterintuitive result that a number
12315:
12240:
12235:
12032:
12027:
11861:
11328:
11176:
1481:{\displaystyle 10^{10^{100}}=(10\uparrow )^{2}100=(10\uparrow )^{3}2}
1231:{\displaystyle 10^{10^{10^{10^{10^{4.829}}}}}=(10\uparrow )^{5}4.829}
1016:{\displaystyle 10\uparrow \uparrow n=10\to n\to 2=(10\uparrow )^{n}1}
849:
403:
388:
255:
251:
232:
177:
159:
133:
886:
12279:
12245:
12220:
11966:
11298:
11152:
11097:
function Ξ£ is an example of a function which grows faster than any
4938:
4909:
10 = 1,000,000,000,000,000 = 1 million billion = 1 quadrillion
366:
137:
12170:
9404:{\displaystyle 1-{\frac {10^{9}}{10^{10}}}=1-{\frac {1}{10}}=90\%}
9269:
In the case of an approximation of an extremely large number, the
11856:
7434:
2913:
An advantage of the first is that when considered as function of
204:
98:
57:
11790:
1119:
More precisely, numbers in between can be expressed in the form
457:". The public's association of the phrase and Sagan came from a
399:
provides a precise asymptotic expression for this rapid growth.
12111:
11916:
11851:
2651:{\displaystyle 10\uparrow \uparrow \uparrow (7.3\times 10^{6})}
1953:{\displaystyle 10^{(10\uparrow )^{n}x}=(10\uparrow )^{n}10^{x}}
848:, larger than what can be represented even using power towers (
488:
203:
The mass of Earth consists of about 4 Γ 10, or 4 sexdecillion,
144:
11370:
3692:{\displaystyle 10\uparrow ^{10\uparrow ^{3\times 10^{5}}10}10}
1812:
to get a number between 1 and 10. Thus, the number is between
11770:"[July 2nd 2024] We have proved "BB(5) = 47,176,870""
9801:
Such comparisons of iterated logarithms are common, e.g., in
2544:{\displaystyle (10\uparrow \uparrow )^{3}(2.8\times 10^{12})}
7737:{\displaystyle 100\uparrow \uparrow 3=10^{2\times 10^{200}}}
4148:
is too large to give exactly, it is possible to use a fixed
101:βa 1 followed by nine zeros: 1,000,000,000. Conversely, the
11111:
4746:
the most significant number, but with decreasing order for
4647:) is very similar to going from the latter to the sequence
4472:
decreases, and with as inner argument a sequence of powers
341:{\displaystyle 10^{10^{10^{10^{10^{1.1}}}}}{\mbox{ years}}}
210:
449:
distinguish between "millions" and "billions", astronomer
11841:
10187:β₯1), it is possible to produce a faster-growing sequence
5203:, the 51st and as of January 2021 the largest known
858:
is a sequence relating to graphs. TREE(3) is larger than
387:
processes give rise to astonishingly large numbers. The
174:
129:
11373:"An estimation of the number of cells in the human body"
10137:
Systematically creating ever-faster-increasing sequences
4812:
is a large number same techniques can be applied again.
60:, they can also take other forms in different contexts.
11435:
10261:). This can be repeated any number of times by letting
9713:{\displaystyle \log _{10}(\log _{10}(10^{10^{10}}))=10}
7256:{\displaystyle 10\uparrow ^{10\uparrow ^{10^{10}}10}10}
3123:{\displaystyle 10\uparrow ^{3}3=(10\uparrow ^{2})^{3}1}
2068:{\displaystyle 10\uparrow \uparrow (7.21\times 10^{8})}
412:, along with similar representations of bit-strings in
11704:{\displaystyle 10\uparrow ^{n}10<3\uparrow ^{n+1}3}
10147:
Given a strictly increasing integer sequence/function
7365:) is between (10 β 10 β 63 β 2) and (10 β 10 β 64 β 2)
7346:) is between (10 β 10 β 10 β 2) and (10 β 10 β 11 β 2)
7173:
The second term in the definition of Graham's number,
6630:{\displaystyle 4\uparrow \uparrow \uparrow \uparrow 4}
1597:
1513:
332:
27:
Numbers significantly larger than those used regularly
11659:
11213:
11185:
10394:
10336:
10267:
10193:
10153:
10096:
10056:
10012:
9992:
9874:
9791:{\displaystyle \log _{10}(\log _{10}(10^{10^{9}}))=9}
9726:
9648:
9607:
9573:
9533:
9499:
9466:
9439:
9342:
9309:
9282:
9239:
9197:
9166:
9058:
8963:
8885:
8809:
8737:
8667:
8639:
8613:
8509:
8340:
8232:
8136:
8108:
8082:
7922:
7751:
7698:
7658:
7571:
7509:
7467:
7327:) is between (10 β 10 β 9 β 2) and (10 β 10 β 10 β 2)
7216:
7189:
7137:
7110:
7032:
6976:
6920:
6864:
6811:
6758:
6705:
6643:
6617:
6564:
6545:
The first term in the definition of Graham's number,
6485:
6432:
6379:
6326:
6273:
6220:
6142:
6089:
6013:
5973:
5920:
5794:
5675:
5596:
5413:
5353:
5221:
5091:
4952:
4847:
4760:
4695:
4653:
4616:
4585:
4539:
4478:
4431:
4358:
4285:
4229:
4174:
4118:
4047:
3951:
3866:
3805:
3765:
3709:
3646:
3613:
3582:
3514:
3422:
3352:
3291:
3256:
3227:
3198:
3144:
3071:
3030:
2976:
2927:
2858:
2817:
2782:
2750:
2667:
2619:
2590:
2560:
2499:
2470:
2450:
2421:
2392:
2362:
2336:
2203:
2175:
2149:
2087:
2036:
1973:
1888:
1844:
1818:
1785:
1745:
1713:
1499:
1410:
1371:
1296:
1270:
1244:
1164:
1125:
1069:
1029:
960:
808:
767:
715:
665:
616:
589:
556:
529:
497:
292:
147:
in the human brain (estimated at 10), or 100 trillion
9191:
changes the result by a factor 10. In a number like
438:
224:β (estimated at around 10), or 1 novemtrigintillion
4579:. Note that the process of going from the sequence
2613:it is possible use the triple arrow operator, e.g.
1112:These are very round numbers, each representing an
11703:
11653:Regarding the comparison with the previous value:
11436:Landenmark HK, Forgan DH, Cockell CS (June 2015).
11236:
11195:
11088:
10407:
10380:
10322:
10242:
10175:
10120:
10082:
10038:
9998:
9971:
9790:
9712:
9627:
9593:
9553:
9519:
9479:
9452:
9403:
9322:
9295:
9252:
9225:
9179:
9139:
9044:
8947:
8871:
8793:
8723:
8651:
8625:
8595:
8495:
8325:
8217:
8120:
8094:
8064:
7907:
7736:
7683:
7637:
7555:
7493:
7255:
7202:
7163:
7123:
7081:
7016:
6960:
6904:
6848:
6795:
6742:
6690:
6629:
6601:
6534:
6469:
6416:
6363:
6310:
6257:
6205:
6126:
6074:
5998:
5957:
5879:
5779:
5660:
5581:
5398:
5328:
5195:
5067:
4874:
4796:
4730:
4677:
4635:
4598:
4563:
4514:
4460:
4413:
4312:
4251:
4211:
4140:
4097:
4033:
3935:
3840:
3787:
3743:
3691:
3632:
3595:
3557:
3488:
3408:
3331:
3277:
3242:
3213:
3180:
3122:
3054:
3008:
2959:
2877:
2830:
2800:
2768:
2736:
2650:
2605:
2575:
2543:
2485:
2456:
2436:
2407:
2374:
2348:
2322:
2187:
2161:
2135:
2067:
2006:the power tower can be made one higher, replacing
1998:
1952:
1868:
1830:
1804:
1763:
1731:
1699:
1480:
1389:
1357:
1282:
1256:
1230:
1150:
1097:
1051:
1015:
835:
794:
748:
696:
629:
602:
569:
542:
510:
340:
11568:. Universe Today. 30-07-2009. Retrieved 02-03-13.
9894:
9822:, but "at a glance" given the decimal expansion.
9204:
7095:( 3 β 2 β 2 β 2 ) = ( 3 β 2 β 9 ) = ( 3 β 3 β 8 )
6039:
3558:{\displaystyle 10\uparrow ^{n}10=(10\to 10\to n)}
3285:is increased by 1 and everything to the right of
2100:
876:
453:stressed the "b". Sagan never did, however, say "
12369:
11528:"XXII. Programming a Computer for Playing Chess"
11360:Β«The study of large numbers is called googologyΒ»
11109:= 1, 2, 3, 4, 5 are 1, 4, 6, 13, 4098 (sequence
9838:yields indistinguishibility firstly between log(
5399:{\displaystyle 10^{10^{100}}=(10\uparrow )^{3}2}
4754:; as inner argument yields a sequence of powers
749:{\displaystyle 10^{\text{googol}}=10^{10^{100}}}
184:on Earth, as a possible approximation of global
11522:
11438:"An Estimate of the Total DNA in the Biosphere"
11264:List of arbitrary-precision arithmetic software
8626:{\displaystyle 100\uparrow \uparrow \uparrow n}
7684:{\displaystyle 100\uparrow \uparrow 2=10^{200}}
3250:is too large to be given exactly, the value of
357:which corresponds to the scale of an estimated
11480:
8652:{\displaystyle 10\uparrow \uparrow \uparrow n}
3572:The above can be applied recursively for this
2375:{\displaystyle 10\uparrow \uparrow \uparrow 5}
2349:{\displaystyle 10\uparrow \uparrow \uparrow 4}
2188:{\displaystyle 10\uparrow \uparrow \uparrow 3}
2162:{\displaystyle 10\uparrow \uparrow \uparrow 2}
2021:from the lower-tower representation of the log
56:. While they often manifest as large positive
12186:
11806:
11119:). Ξ£(6) is not known but is at least 10ββ15.
7556:{\displaystyle 100^{100^{12}}=10^{2*10^{24}}}
7037:↑ ↑ ↑ ↑ ↑
11637:"GIMPS Discovers Largest Known Prime Number"
7454:
7407:Some notations for extremely large numbers:
7183:( 10 β 10 β 3 β 2 ) = (10 β 10 β (10 β 10 β
5999:{\displaystyle (10\uparrow \uparrow )^{2}11}
4916:Numbers expressible in scientific notation:
4223:, allowing us to write a number in the form
3759:, allowing us to write a number in the form
3755:), these levels become functional powers of
11237:{\displaystyle {\mathfrak {c}}=\aleph _{1}}
4906:3 = 3 ββ 3 = 7,625,597,484,987 β 7.63 Γ 10
4098:{\displaystyle G<f^{64}(4)<f^{65}(1)}
2386:Similarly to the above, if the exponent of
915:. Unsourced material may be challenged and
12193:
12179:
11813:
11799:
11579:Heat Kernel Techniques and Quantum Gravity
4922:number of atoms in the observable universe
1764:{\displaystyle 10\uparrow \uparrow 65,534}
1732:{\displaystyle 10\uparrow \uparrow 65,533}
954:Tetration with base 10 gives the sequence
158:; the number of atoms in 12 grams of
136:(as of 2024, typically about 10, 1β2
11463:
11453:
11388:
11349:One Million Things: A Visual Encyclopedia
9893:
9857:
9264:
9226:{\displaystyle 10^{\,\!6.2\times 10^{3}}}
9203:
6038:
4820:Numbers expressible in decimal notation:
4797:{\displaystyle (10\uparrow ^{n})^{p_{n}}}
4515:{\displaystyle (10\uparrow ^{n})^{p_{n}}}
4219:these levels become functional powers of
3181:{\displaystyle (10\uparrow ^{n})^{k_{n}}}
2099:
1869:{\displaystyle 10\uparrow \uparrow (n+1)}
1550:
1544:
1538:
935:Learn how and when to remove this message
11151:. Certain fields of mathematics define
7501:, with base 10 the exponent is doubled.
4903:10 = 1,000,000,000,000 = 1 trillion
4156:= 1, and apply the above recursively to
1023:, the power towers of numbers 10, where
67:
10381:{\displaystyle f_{\omega }(n)=f_{n}(n)}
10323:{\displaystyle f_{k}(n)=f_{k-1}^{n}(n)}
9830:becomes humanly indistinguishable from
9151:is large this is "approximately" equal)
7164:{\displaystyle 10\uparrow ^{10^{10}}10}
4944:5 = 5 ββ 3 = 5 β 1.91 Γ 10 β (10 β) 3.3
4937:composing the volume of the observable
4930:4 = 4 ββ 3 = 2 β 1.34 Γ 10 β (10 β) 2.2
4414:{\displaystyle f_{k}(n)=f_{k-1}^{n}(1)}
14:
12370:
11179:is the next greatest cardinal number.
8095:{\displaystyle 100\uparrow \uparrow n}
3744:{\displaystyle f(n)=10\uparrow ^{n}10}
3192:(with exactly given integer exponents
3009:{\displaystyle (10\uparrow ^{2})^{3}b}
2606:{\displaystyle (10\uparrow \uparrow )}
2576:{\displaystyle (10\uparrow \uparrow )}
2486:{\displaystyle (10\uparrow \uparrow )}
1399:extension of tetration to real heights
12174:
11794:
11571:
11486:
10243:{\displaystyle f_{1}(n)=f_{0}^{n}(n)}
9422:magnify relative errors greatly β if
8121:{\displaystyle 10\uparrow \uparrow n}
7101:( 10 β 2 β 2 β 2 ) = ( 10 β 2 β 100 )
3841:{\displaystyle f^{2}(3\times 10^{5})}
2960:{\displaystyle (a\uparrow ^{n})^{k}b}
1831:{\displaystyle 10\uparrow \uparrow n}
1283:{\displaystyle 10\uparrow \uparrow 6}
1257:{\displaystyle 10\uparrow \uparrow 5}
11641:Great Internet Mersenne Prime Search
9834:(taking iterated logarithms of such
9063:↑ ↑ ↑ ↑
8968:↑ ↑ ↑ ↑
8890:↑ ↑ ↑ ↑
8814:↑ ↑ ↑ ↑
8742:↑ ↑ ↑ ↑
8672:↑ ↑ ↑ ↑
7049:↑ ↑ ↑ ↑
6981:↑ ↑ ↑ ↑
6925:↑ ↑ ↑ ↑
6869:↑ ↑ ↑ ↑
6816:↑ ↑ ↑ ↑
6763:↑ ↑ ↑ ↑
6710:↑ ↑ ↑ ↑
6622:↑ ↑ ↑ ↑
6569:↑ ↑ ↑ ↑
6490:↑ ↑ ↑ ↑
913:adding citations to reliable sources
880:
121:in the human body (estimated at 3.72
12200:
11216:
11188:
11122:
10388:, which grows much faster than any
10083:{\displaystyle 2^{n}\approx 10^{n}}
10039:{\displaystyle n^{n}\approx 10^{n}}
8607:is large it seems fair to say that
8076:is large it seems fair to say that
5210:9 = 9 ββ 3 β 4.28 Γ 10 β (10 β) 8.6
5083:8 = 8 ββ 3 β 6.01 Γ 10 β (10 β) 7.2
5080:7 = 7 ββ 3 β 3.76 Γ 10 β (10 β) 5.8
5077:6 = 6 ββ 3 β 2.66 Γ 10 β (10 β) 4.6
4897:10 = 1,000,000,000 = 1 billion
637:, depending on number naming system
577:, depending on number naming system
379:properties of statistical mechanics
24:
11225:
9490:the relative error is larger, and
9398:
7402:
7098:( 10 β 10 β 10 ) = ( 10 β 2 β 11 )
7089:= ( 10 β 2 β 5 ) = ( 10 β 10 β 4 )
6542:= ( 10 β 2 β 4 ) = ( 10 β 10 β 3 )
1999:{\displaystyle (10\uparrow )^{n}x}
1967:is too large for a representation
1151:{\displaystyle (10\uparrow )^{n}a}
836:{\displaystyle 10^{10^{10^{964}}}}
25:
12394:
12087:Indefinite and fictitious numbers
11820:
11289:Indefinite and fictitious numbers
7104:( 10 β 10 β 2 β 2 ) = ( 10 β 2 β
7092:( 2 β 3 β 2 β 2 ) = ( 2 β 3 β 8 )
4731:{\displaystyle {f_{qk}}^{m_{qk}}}
4636:{\displaystyle 10\uparrow ^{n}10}
3332:{\displaystyle ({n+1})^{k_{n+1}}}
1052:{\displaystyle (10\uparrow )^{n}}
795:{\displaystyle 10^{10^{10^{34}}}}
162: – approximately
11746:"Large Numbers (page 2) at MROB"
7494:{\displaystyle 100^{12}=10^{24}}
3500:can be so large that, in a way,
885:
697:{\displaystyle 2^{82,589,933}-1}
463:skit. Parodying Sagan's effect,
11762:
11738:
11714:
11647:
11629:
11611:
11492:"Counting All the DNA on Earth"
11196:{\displaystyle {\mathfrak {c}}}
11089:In some noncomputable sequences
7395:( 10 β 10 β 10 β 10 β 10 β 10 )
4461:{\displaystyle {f_{k}}^{m_{k}}}
2878:{\displaystyle a\uparrow ^{n}b}
2841:Compare this notation with the
1577:
1576:
266:spans an incredible 93 billion
245:
11683:
11664:
11600:
11559:
11516:
11429:
11364:
11353:
11342:
11269:Dirac large numbers hypothesis
11259:Arbitrary-precision arithmetic
11143:totally indescribable cardinal
10419:(here Ο is the first infinite
10375:
10369:
10353:
10347:
10317:
10311:
10284:
10278:
10237:
10231:
10210:
10204:
10170:
10164:
10112:
10100:
9889:
9875:
9779:
9776:
9756:
9740:
9701:
9698:
9678:
9662:
9125:
9121:
9115:
9106:
9102:
9096:
9081:
9077:
9071:
9062:
9030:
9026:
9020:
9011:
9007:
9001:
8986:
8982:
8976:
8967:
8933:
8929:
8923:
8914:
8910:
8904:
8901:
8889:
8857:
8853:
8847:
8838:
8834:
8828:
8825:
8813:
8779:
8775:
8769:
8760:
8756:
8750:
8741:
8709:
8705:
8699:
8690:
8686:
8680:
8671:
8643:
8617:
8590:
8578:
8575:
8557:
8553:
8547:
8532:
8528:
8522:
8513:
8490:
8478:
8475:
8457:
8453:
8447:
8432:
8428:
8422:
8416:
8397:
8388:
8384:
8378:
8363:
8359:
8353:
8344:
8311:
8307:
8301:
8298:
8289:
8270:
8261:
8257:
8251:
8248:
8236:
8203:
8199:
8193:
8187:
8168:
8159:
8155:
8149:
8140:
8112:
8086:
8059:
8047:
8044:
8026:
8022:
8016:
7995:
7991:
7985:
7979:
7960:
7945:
7941:
7935:
7926:
7893:
7889:
7883:
7868:
7864:
7858:
7852:
7833:
7824:
7820:
7814:
7808:
7783:
7774:
7770:
7764:
7755:
7702:
7662:
7229:
7221:
7142:
7067:
7063:
7057:
7048:
7036:
6999:
6995:
6989:
6980:
6943:
6939:
6933:
6924:
6887:
6883:
6877:
6868:
6834:
6830:
6824:
6815:
6781:
6777:
6771:
6762:
6728:
6724:
6718:
6709:
6676:
6672:
6666:
6657:
6653:
6647:
6621:
6587:
6583:
6577:
6568:
6520:
6516:
6510:
6501:
6489:
6455:
6451:
6445:
6436:
6402:
6398:
6392:
6383:
6349:
6345:
6339:
6330:
6296:
6292:
6286:
6277:
6243:
6239:
6233:
6224:
6200:
6181:
6172:
6168:
6162:
6153:
6149:
6143:
6112:
6108:
6102:
6093:
6024:
6020:
6014:
5984:
5980:
5974:
5943:
5939:
5933:
5924:
5901:10 βββ 2 = 10 ββ 10 = (10 β) 1
5865:
5861:
5855:
5846:
5765:
5761:
5755:
5713:
5646:
5642:
5636:
5627:
5567:
5563:
5557:
5540:
5536:
5530:
5508:
5504:
5498:
5458:
5384:
5380:
5374:
5339:Numbers expressible in (10 β)
5314:
5310:
5304:
5252:
5178:
5174:
5168:
5053:
5049:
5043:
4990:
4778:
4768:
4761:
4672:
4666:
4621:
4558:
4552:
4496:
4486:
4479:
4408:
4402:
4375:
4369:
4307:
4301:
4246:
4240:
4206:
4200:
4184:
4178:
4164:this gives multiple levels of
4135:
4129:
4092:
4086:
4070:
4064:
4028:
4022:
4006:
4000:
3994:
3988:
3982:
3973:
3967:
3961:
3909:
3901:
3891:
3885:
3879:
3873:
3867:
3835:
3816:
3782:
3776:
3729:
3719:
3713:
3659:
3651:
3633:{\displaystyle 3\times 10^{5}}
3584:
3552:
3546:
3540:
3534:
3519:
3480:
3471:
3459:
3453:
3450:
3441:
3435:
3429:
3426:
3394:
3390:
3384:
3369:
3365:
3359:
3356:
3307:
3292:
3162:
3152:
3145:
3108:
3098:
3091:
3076:
2994:
2984:
2977:
2945:
2935:
2928:
2863:
2819:
2792:
2786:
2760:
2754:
2731:
2712:
2703:
2699:
2693:
2684:
2680:
2674:
2671:
2645:
2626:
2623:
2600:
2597:
2591:
2570:
2567:
2561:
2538:
2519:
2510:
2506:
2500:
2480:
2477:
2471:
2431:
2428:
2422:
2402:
2399:
2393:
2366:
2340:
2317:
2298:
2289:
2285:
2279:
2270:
2266:
2260:
2254:
2235:
2226:
2222:
2216:
2213:
2207:
2179:
2153:
2091:
2062:
2043:
2040:
1984:
1980:
1974:
1931:
1927:
1921:
1904:
1900:
1894:
1863:
1851:
1848:
1822:
1749:
1717:
1679:
1675:
1669:
1663:
1638:
1623:
1619:
1613:
1503:
1466:
1462:
1456:
1441:
1437:
1431:
1352:
1340:
1337:
1319:
1315:
1309:
1300:
1274:
1248:
1216:
1212:
1206:
1136:
1132:
1126:
1079:
1073:
1040:
1036:
1030:
1001:
997:
991:
982:
976:
964:
877:Standardized system of writing
414:algorithmic information theory
377:arbitrarily many times due to
13:
1:
12342:Conway chained arrow notation
12002:Conway chained arrow notation
11335:
11309:Myriads (10,000) in East Asia
10767:Conway chained arrow notation
9430:have a small relative error,
7426:Conway chained arrow notation
4875:{\displaystyle 2^{2^{2^{2}}}}
4815:
4212:{\displaystyle g(n)=f^{n}(1)}
3596:{\displaystyle \uparrow ^{n}}
3346:are not needed. For example,
2847:Conway chained arrow notation
2831:{\displaystyle \uparrow ^{n}}
2437:{\displaystyle (10\uparrow )}
2408:{\displaystyle (10\uparrow )}
761:: the first is approximately
217:(10), or 100 quinvigintillion
32:Large number (disambiguation)
11455:10.1371/journal.pbio.1002168
11390:10.3109/03014460.2013.807878
10426:For example, starting with
9846:), secondly between log(log(
9594:{\displaystyle 10^{10^{10}}}
8633:is "approximately equal to"
8102:is "approximately equal to"
4313:{\displaystyle f_{k}^{m}(n)}
1390:{\displaystyle 1<a<10}
7:
11251:
10669:in the sequence defined by
9628:{\displaystyle 10^{10^{9}}}
9554:{\displaystyle 10^{10^{b}}}
9520:{\displaystyle 10^{10^{a}}}
9155:
9078:↑ ↑ ↑
8983:↑ ↑ ↑
8902:↑ ↑ ↑
8826:↑ ↑ ↑
8644:↑ ↑ ↑
8618:↑ ↑ ↑
8576:↑ ↑ ↑
8514:↑ ↑ ↑
8476:↑ ↑ ↑
8345:↑ ↑ ↑
8237:↑ ↑ ↑
8141:↑ ↑ ↑
7308:β 1) > (10 β 10 β 4 β 2)
7285:β 1) > (10 β 10 β 3 β 2)
7064:↑ ↑ ↑
6996:↑ ↑ ↑
6940:↑ ↑ ↑
6884:↑ ↑ ↑
6831:↑ ↑ ↑
6778:↑ ↑ ↑
6725:↑ ↑ ↑
6654:↑ ↑ ↑
6584:↑ ↑ ↑
6502:↑ ↑ ↑
6437:↑ ↑ ↑
6384:↑ ↑ ↑
6331:↑ ↑ ↑
6278:↑ ↑ ↑
6225:↑ ↑ ↑
6094:↑ ↑ ↑
5925:↑ ↑ ↑
5213:10 =10 ββ 3 = 10 = (10 β) 1
4678:{\displaystyle {f_{n}}(10)}
4564:{\displaystyle {f_{n}}(10)}
3481:↑ ↑ ↑
3436:↑ ↑ ↑
2793:↑ ↑ ↑
2787:↑ ↑ ↑
2761:↑ ↑ ↑
2755:↑ ↑ ↑
2672:↑ ↑ ↑
2624:↑ ↑ ↑
2367:↑ ↑ ↑
2341:↑ ↑ ↑
2180:↑ ↑ ↑
2154:↑ ↑ ↑
1504:↑ ↑ ↑
1098:{\displaystyle f(n)=10^{n}}
482:
391:function, which quantifies
10:
12399:
12294:Inverse for right argument
12097:Largest known prime number
11623:December 21, 2016, at the
11132:
11126:
10704:> 3 β 3 + 2, and hence
10140:
9808:
7392:( 10 β 10 β 10 β 10 β 10 )
4888:10 = 1,000,000 = 1 million
4804:with decreasing values of
4522:with decreasing values of
3188:with decreasing values of
71:
29:
12352:Knuth's up-arrow notation
12329:
12293:
12254:
12208:
12150:
12082:Extended real number line
12062:
12015:
11997:Knuth's up-arrow notation
11984:
11975:
11828:
11774:The Busy Beaver Challenge
7455:Comparison of base values
7412:Knuth's up-arrow notation
6554:Graham's number#Magnitude
4882:= 2 ββ 4 = 2βββ3 = 65,536
4168:. Introducing a function
3278:{\displaystyle {k_{n+1}}}
630:{\displaystyle 10^{6000}}
603:{\displaystyle 10^{3003}}
433:RobertsonβSeymour theorem
12357:SteinhausβMoser notation
12007:SteinhausβMoser notation
11279:History of large numbers
11207:. The proposition that
11205:cardinality of the reals
10176:{\displaystyle f_{0}(n)}
7431:Steinhaus-Moser notation
4252:{\displaystyle g^{m}(n)}
4141:{\displaystyle f^{m}(n)}
3788:{\displaystyle f^{m}(n)}
2444:, it is possible to add
1805:{\displaystyle log_{10}}
570:{\displaystyle 10^{600}}
543:{\displaystyle 10^{303}}
511:{\displaystyle 10^{100}}
359:PoincarΓ© recurrence time
209:The estimated number of
11722:"Large Numbers at MROB"
11607:How to Get A Googolplex
11377:Annals of Human Biology
10689:This follows by noting
10511:Knuth up-arrow notation
10250:(where the superscript
10048:the computation of mega
9296:{\displaystyle 10^{10}}
9253:{\displaystyle 10^{50}}
7203:{\displaystyle 10^{10}}
7124:{\displaystyle 10^{10}}
4468:where going inward the
4344:, etc., obtains (10β10β
3243:{\displaystyle {k_{n}}}
3214:{\displaystyle {k_{n}}}
3055:{\displaystyle ''b''=1}
3016:= ( 10 β ( 10 β ( 10 β
439:"Billions and billions"
250:In the vast expanse of
12050:Fast-growing hierarchy
11705:
11535:Philosophical Magazine
11314:Names of large numbers
11238:
11197:
10409:
10382:
10324:
10244:
10177:
10143:fast-growing hierarchy
10122:
10084:
10040:
10000:
9973:
9858:Approximate arithmetic
9803:analytic number theory
9792:
9714:
9629:
9595:
9555:
9521:
9481:
9480:{\displaystyle 10^{b}}
9454:
9453:{\displaystyle 10^{a}}
9405:
9333:The relative error is
9324:
9323:{\displaystyle 10^{9}}
9297:
9265:For very large numbers
9254:
9227:
9181:
9180:{\displaystyle 10^{n}}
9141:
9046:
8949:
8873:
8795:
8725:
8653:
8627:
8597:
8497:
8327:
8219:
8122:
8096:
8066:
7909:
7738:
7685:
7639:
7557:
7495:
7441:Fast-growing hierarchy
7257:
7204:
7180:= 3 β 3 > 10 β 10.
7165:
7125:
7083:
7018:
6962:
6906:
6850:
6797:
6744:
6692:
6631:
6603:
6536:
6471:
6418:
6365:
6312:
6259:
6207:
6128:
6076:
6000:
5959:
5881:
5781:
5662:
5583:
5400:
5330:
5197:
5069:
4933:Approximate number of
4876:
4798:
4732:
4679:
4637:
4600:
4599:{\displaystyle 10^{n}}
4575:) with an approximate
4565:
4516:
4462:
4415:
4314:
4253:
4213:
4142:
4099:
4035:
3937:
3842:
3789:
3745:
3693:
3634:
3607:(10 β 10 β (10 β 10 β
3597:
3559:
3490:
3410:
3333:
3279:
3244:
3215:
3182:
3124:
3056:
3010:
2961:
2879:
2832:
2802:
2770:
2738:
2652:
2607:
2577:
2545:
2487:
2458:
2438:
2409:
2376:
2350:
2324:
2189:
2163:
2137:
2069:
2000:
1954:
1870:
1832:
1806:
1779:) one has to take the
1765:
1733:
1701:
1482:
1391:
1359:
1284:
1258:
1232:
1152:
1099:
1053:
1017:
856:Kruskal's tree theorem
837:
796:
750:
698:
631:
604:
571:
544:
512:
429:Kruskalβs tree theorem
342:
200:, or 53Β±36 undecillion
170:, or 602.2 sextillion.
86:Names of large numbers
12383:Mathematical notation
12347:Grzegorczyk hierarchy
12107:Long and short scales
12045:Grzegorczyk hierarchy
11706:
11566:Atoms in the Universe
11541:(314). Archived from
11239:
11198:
10410:
10408:{\displaystyle f_{k}}
10383:
10325:
10245:
10178:
10123:
10085:
10041:
10001:
9974:
9793:
9715:
9630:
9596:
9556:
9522:
9482:
9455:
9420:exponential functions
9406:
9325:
9298:
9255:
9228:
9187:, one unit change in
9182:
9142:
9047:
8950:
8874:
8796:
8726:
8654:
8628:
8598:
8498:
8328:
8220:
8123:
8097:
8067:
7910:
7739:
7686:
7640:
7558:
7496:
7422:, including tetration
7389:( 10 β 10 β 10 β 10 )
7258:
7205:
7166:
7126:
7084:
7019:
6963:
6907:
6851:
6798:
6745:
6693:
6632:
6604:
6537:
6472:
6419:
6366:
6313:
6260:
6208:
6129:
6077:
6001:
5960:
5882:
5782:
5663:
5584:
5401:
5331:
5198:
5070:
4877:
4799:
4740:lexicographical order
4733:
4680:
4638:
4601:
4566:
4517:
4463:
4416:
4315:
4254:
4214:
4143:
4100:
4036:
3938:
3843:
3790:
3746:
3694:
3635:
3598:
3560:
3491:
3411:
3334:
3280:
3245:
3216:
3183:
3125:
3057:
3011:
2962:
2905: + 2,
2880:
2833:
2803:
2771:
2739:
2653:
2608:
2578:
2546:
2488:
2459:
2439:
2410:
2377:
2351:
2325:
2190:
2164:
2138:
2070:
2001:
1955:
1871:
1833:
1807:
1766:
1734:
1702:
1599: copies of
1483:
1392:
1360:
1285:
1259:
1233:
1153:
1100:
1054:
1018:
838:
797:
751:
699:
632:
605:
572:
545:
513:
455:billions and billions
343:
125:10), or 37.2 trillion
68:In the everyday world
54:statistical mechanics
11657:
11304:Law of large numbers
11246:continuum hypothesis
11211:
11183:
10801:-1 β 2) (because if
10392:
10334:
10265:
10191:
10151:
10094:
10054:
10010:
9990:
9872:
9724:
9646:
9605:
9571:
9531:
9497:
9464:
9437:
9340:
9307:
9280:
9237:
9195:
9164:
9056:
8961:
8883:
8807:
8735:
8665:
8637:
8611:
8507:
8338:
8230:
8134:
8106:
8080:
7920:
7749:
7696:
7656:
7569:
7507:
7465:
7386:( 10 β 10 β 10 β 4 )
7383:( 10 β 10 β 10 β 3 )
7380:( 10 β 10 β 65 β 2 )
7368:( 10 β 10 β 64 β 2 )
7330:( 10 β 10 β 10 β 2 )
7214:
7187:
7135:
7108:
7030:
6974:
6918:
6862:
6809:
6756:
6703:
6641:
6615:
6562:
6483:
6430:
6377:
6324:
6271:
6218:
6140:
6087:
6011:
5971:
5918:
5892:3 ββ 6 β (10 β) 1.10
5792:
5673:
5594:
5411:
5351:
5219:
5089:
4950:
4845:
4758:
4693:
4651:
4614:
4583:
4537:
4476:
4429:
4356:
4283:
4227:
4172:
4116:
4045:
3949:
3864:
3803:
3763:
3707:
3644:
3611:
3580:
3565:with an approximate
3512:
3420:
3350:
3289:
3254:
3225:
3196:
3142:
3069:
3028:
2974:
2925:
2856:
2815:
2780:
2748:
2665:
2617:
2588:
2558:
2497:
2468:
2448:
2419:
2390:
2360:
2334:
2201:
2173:
2147:
2085:
2034:
1971:
1886:
1842:
1816:
1783:
1743:
1711:
1497:
1408:
1369:
1294:
1268:
1242:
1162:
1123:
1067:
1027:
958:
909:improve this section
806:
765:
713:
663:
614:
587:
554:
527:
495:
290:
173:The total number of
145:neuronal connections
30:For other uses, see
12321:Super-logarithm (4)
12280:Root extraction (3)
12122:Orders of magnitude
11992:Scientific notation
11157:transfinite numbers
10642:is a unary version)
10310:
10230:
5895:2 ββ 8 β (10 β) 4.3
4900:10 = 10,000,000,000
4401:
4300:
2554:If the exponent of
2464:to the exponent of
1566:
1404:Thus googolplex is
1238:, a number between
872:on 26 January 2007.
443:To help viewers of
264:observable universe
215:observable universe
97:10, we express one
91:Scientific notation
82:Orders of magnitude
74:Scientific notation
12337:Ackermann function
12231:Exponentiation (3)
12226:Multiplication (2)
12040:Ackermann function
11701:
11496:The New York Times
11274:Exponential growth
11234:
11193:
10633:Ackermann function
10405:
10378:
10320:
10290:
10240:
10216:
10173:
10118:
10113:↑ ↑
10101:↑ ↑
10080:
10036:
9996:
9969:
9788:
9710:
9625:
9591:
9551:
9517:
9477:
9450:
9418:The point is that
9401:
9320:
9293:
9250:
9223:
9177:
9137:
9103:↑ ↑
9042:
9008:↑ ↑
8945:
8911:↑ ↑
8869:
8835:↑ ↑
8791:
8757:↑ ↑
8721:
8687:↑ ↑
8649:
8623:
8593:
8529:↑ ↑
8493:
8429:↑ ↑
8360:↑ ↑
8323:
8299:↑ ↑
8249:↑ ↑
8215:
8118:
8113:↑ ↑
8092:
8087:↑ ↑
8062:
8045:↑ ↑
7927:↑ ↑
7905:
7756:↑ ↑
7734:
7703:↑ ↑
7681:
7663:↑ ↑
7635:
7553:
7491:
7420:Ackermann function
7301:) > (10 β 10 β
7278:) > (10 β 10 β
7253:
7200:
7161:
7121:
7079:
7014:
6958:
6902:
6846:
6793:
6740:
6688:
6673:↑ ↑
6627:
6599:
6532:
6517:↑ ↑
6467:
6452:↑ ↑
6414:
6399:↑ ↑
6361:
6346:↑ ↑
6308:
6293:↑ ↑
6255:
6240:↑ ↑
6203:
6150:↑ ↑
6124:
6109:↑ ↑
6072:
6021:↑ ↑
5996:
5981:↑ ↑
5955:
5940:↑ ↑
5898:10 ββ 6 = (10 β) 1
5889:10 ββ 5 = (10 β) 1
5877:
5847:↑ ↑
5777:
5714:↑ ↑
5658:
5628:↑ ↑
5579:
5459:↑ ↑
5396:
5326:
5253:↑ ↑
5193:
5065:
4991:↑ ↑
4872:
4794:
4728:
4675:
4633:
4610:) to the sequence
4596:
4561:
4512:
4458:
4411:
4381:
4310:
4286:
4249:
4209:
4138:
4095:
4031:
3933:
3860:β2). For example,
3838:
3785:
3741:
3689:
3630:
3593:
3576:, so the notation
3555:
3486:
3460:↑ ↑
3451:↑ ↑
3406:
3391:↑ ↑
3366:↑ ↑
3329:
3275:
3240:
3211:
3178:
3120:
3052:
3006:
2957:
2875:
2828:
2798:
2766:
2734:
2681:↑ ↑
2648:
2603:
2598:↑ ↑
2573:
2568:↑ ↑
2541:
2507:↑ ↑
2483:
2478:↑ ↑
2454:
2434:
2405:
2372:
2346:
2320:
2267:↑ ↑
2214:↑ ↑
2208:↑ ↑
2185:
2159:
2133:
2092:↑ ↑
2065:
2041:↑ ↑
1996:
1963:I.e., if a number
1950:
1866:
1849:↑ ↑
1828:
1823:↑ ↑
1802:
1761:
1750:↑ ↑
1729:
1718:↑ ↑
1697:
1608:
1601:
1570:
1518:
1478:
1387:
1355:
1338:↑ ↑
1301:↑ ↑
1280:
1275:↑ ↑
1254:
1249:↑ ↑
1228:
1148:
1114:order of magnitude
1095:
1049:
1013:
965:↑ ↑
833:
792:
746:
694:
655:The largest known
644:= (10β1) Γ (10 + 3
640:The largest known
627:
600:
567:
540:
508:
397:Stirling's formula
338:
336:
188:, is estimated at
180:within the entire
12365:
12364:
12258:for left argument
12168:
12167:
12058:
12057:
10509:for n β₯ 2 (using
9999:{\displaystyle n}
9850:)) and 1+log(log(
9390:
9371:
7371:Graham's number,
5189:
3020:β 2 ) β 2 ) β 2 )
2493:, to obtain e.g.
2457:{\displaystyle 1}
1600:
1586:
1583:
1580:
1517:
1490:Another example:
1107:the googol family
945:
944:
937:
723:
369:whose mass is 10
335:
152:Avogadro constant
140:), or 10 trillion
78:Logarithmic scale
16:(Redirected from
12390:
12330:Related articles
12195:
12188:
12181:
12172:
12171:
11982:
11981:
11912:Eddington number
11857:Hundred thousand
11815:
11808:
11801:
11792:
11791:
11785:
11784:
11782:
11781:
11766:
11760:
11759:
11757:
11756:
11742:
11736:
11735:
11733:
11732:
11718:
11712:
11710:
11708:
11707:
11702:
11697:
11696:
11672:
11671:
11651:
11645:
11644:
11633:
11627:
11615:
11609:
11604:
11598:
11575:
11569:
11563:
11557:
11556:
11554:
11553:
11547:
11532:
11520:
11514:
11513:
11511:
11510:
11490:(18 July 2015).
11484:
11478:
11477:
11467:
11457:
11433:
11427:
11426:
11392:
11368:
11362:
11357:
11351:
11346:
11244:is known as the
11243:
11241:
11240:
11235:
11233:
11232:
11220:
11219:
11202:
11200:
11199:
11194:
11192:
11191:
11123:Infinite numbers
11114:
10414:
10412:
10411:
10406:
10404:
10403:
10387:
10385:
10384:
10379:
10368:
10367:
10346:
10345:
10329:
10327:
10326:
10321:
10309:
10304:
10277:
10276:
10259:functional power
10249:
10247:
10246:
10241:
10229:
10224:
10203:
10202:
10182:
10180:
10179:
10174:
10163:
10162:
10127:
10125:
10124:
10119:
10089:
10087:
10086:
10081:
10079:
10078:
10066:
10065:
10045:
10043:
10042:
10037:
10035:
10034:
10022:
10021:
10005:
10003:
10002:
9997:
9978:
9976:
9975:
9970:
9968:
9967:
9966:
9965:
9958:
9957:
9929:
9928:
9927:
9926:
9906:
9905:
9904:
9903:
9887:
9886:
9797:
9795:
9794:
9789:
9775:
9774:
9773:
9772:
9752:
9751:
9736:
9735:
9719:
9717:
9716:
9711:
9697:
9696:
9695:
9694:
9674:
9673:
9658:
9657:
9634:
9632:
9631:
9626:
9624:
9623:
9622:
9621:
9600:
9598:
9597:
9592:
9590:
9589:
9588:
9587:
9560:
9558:
9557:
9552:
9550:
9549:
9548:
9547:
9526:
9524:
9523:
9518:
9516:
9515:
9514:
9513:
9486:
9484:
9483:
9478:
9476:
9475:
9459:
9457:
9456:
9451:
9449:
9448:
9410:
9408:
9407:
9402:
9391:
9383:
9372:
9370:
9369:
9360:
9359:
9350:
9329:
9327:
9326:
9321:
9319:
9318:
9302:
9300:
9299:
9294:
9292:
9291:
9259:
9257:
9256:
9251:
9249:
9248:
9232:
9230:
9229:
9224:
9222:
9221:
9220:
9219:
9186:
9184:
9183:
9178:
9176:
9175:
9146:
9144:
9143:
9138:
9133:
9132:
9114:
9113:
9095:
9094:
9051:
9049:
9048:
9043:
9038:
9037:
9019:
9018:
9000:
8999:
8954:
8952:
8951:
8946:
8941:
8940:
8922:
8921:
8878:
8876:
8875:
8870:
8865:
8864:
8846:
8845:
8800:
8798:
8797:
8792:
8787:
8786:
8768:
8767:
8730:
8728:
8727:
8722:
8717:
8716:
8698:
8697:
8658:
8656:
8655:
8650:
8632:
8630:
8629:
8624:
8602:
8600:
8599:
8594:
8565:
8564:
8546:
8545:
8502:
8500:
8499:
8494:
8465:
8464:
8446:
8445:
8415:
8414:
8396:
8395:
8377:
8376:
8332:
8330:
8329:
8324:
8319:
8318:
8288:
8287:
8269:
8268:
8224:
8222:
8221:
8216:
8211:
8210:
8186:
8185:
8167:
8166:
8127:
8125:
8124:
8119:
8101:
8099:
8098:
8093:
8071:
8069:
8068:
8063:
8034:
8033:
8009:
8008:
7978:
7977:
7959:
7958:
7914:
7912:
7911:
7906:
7901:
7900:
7876:
7875:
7851:
7850:
7832:
7831:
7801:
7800:
7782:
7781:
7743:
7741:
7740:
7735:
7733:
7732:
7731:
7730:
7690:
7688:
7687:
7682:
7680:
7679:
7644:
7642:
7641:
7636:
7634:
7633:
7632:
7631:
7624:
7623:
7595:
7594:
7593:
7592:
7591:
7590:
7562:
7560:
7559:
7554:
7552:
7551:
7550:
7549:
7526:
7525:
7524:
7523:
7500:
7498:
7497:
7492:
7490:
7489:
7477:
7476:
7262:
7260:
7259:
7254:
7249:
7248:
7244:
7243:
7242:
7241:
7209:
7207:
7206:
7201:
7199:
7198:
7170:
7168:
7167:
7162:
7157:
7156:
7155:
7154:
7130:
7128:
7127:
7122:
7120:
7119:
7088:
7086:
7085:
7080:
7075:
7074:
7024:= ( 10 β 9 β 4 )
7023:
7021:
7020:
7015:
7007:
7006:
6968:= ( 10 β 8 β 4 )
6967:
6965:
6964:
6959:
6951:
6950:
6912:= ( 10 β 7 β 4 )
6911:
6909:
6908:
6903:
6895:
6894:
6856:= ( 10 β 6 β 4 )
6855:
6853:
6852:
6847:
6842:
6841:
6803:= ( 10 β 5 β 4 )
6802:
6800:
6799:
6794:
6789:
6788:
6750:= ( 10 β 4 β 4 )
6749:
6747:
6746:
6741:
6736:
6735:
6697:
6695:
6694:
6689:
6684:
6683:
6665:
6664:
6637:= ( 4 β 4 β 4 )
6636:
6634:
6633:
6628:
6608:
6606:
6605:
6600:
6595:
6594:
6541:
6539:
6538:
6533:
6528:
6527:
6477:= ( 10 β 9 β 3 )
6476:
6474:
6473:
6468:
6463:
6462:
6424:= ( 10 β 8 β 3 )
6423:
6421:
6420:
6415:
6410:
6409:
6371:= ( 10 β 7 β 3 )
6370:
6368:
6367:
6362:
6357:
6356:
6318:= ( 10 β 6 β 3 )
6317:
6315:
6314:
6309:
6304:
6303:
6265:= ( 10 β 5 β 3 )
6264:
6262:
6261:
6256:
6251:
6250:
6212:
6210:
6209:
6204:
6199:
6198:
6180:
6179:
6161:
6160:
6134:= ( 10 β 4 β 3 )
6133:
6131:
6130:
6125:
6120:
6119:
6081:
6079:
6078:
6073:
6071:
6070:
6069:
6068:
6067:
6066:
6065:
6064:
6032:
6031:
6005:
6003:
6002:
5997:
5992:
5991:
5965:= ( 10 β 3 β 3 )
5964:
5962:
5961:
5956:
5951:
5950:
5908:Bigger numbers:
5886:
5884:
5883:
5878:
5873:
5872:
5839:
5838:
5837:
5836:
5835:
5834:
5833:
5832:
5831:
5830:
5829:
5828:
5786:
5784:
5783:
5778:
5773:
5772:
5751:
5750:
5749:
5748:
5747:
5746:
5706:
5705:
5704:
5703:
5702:
5701:
5700:
5699:
5667:
5665:
5664:
5659:
5654:
5653:
5620:
5619:
5618:
5617:
5616:
5615:
5588:
5586:
5585:
5580:
5575:
5574:
5553:
5552:
5548:
5547:
5521:
5520:
5516:
5515:
5489:
5488:
5487:
5486:
5451:
5450:
5449:
5448:
5447:
5446:
5445:
5444:
5443:
5442:
5405:
5403:
5402:
5397:
5392:
5391:
5370:
5369:
5368:
5367:
5335:
5333:
5332:
5327:
5322:
5321:
5300:
5299:
5245:
5244:
5243:
5242:
5241:
5240:
5202:
5200:
5199:
5194:
5187:
5186:
5185:
5164:
5163:
5162:
5161:
5144:
5143:
5113:
5112:
5074:
5072:
5071:
5066:
5061:
5060:
5039:
5038:
5014:
5013:
4983:
4982:
4981:
4980:
4979:
4978:
4977:
4976:
4881:
4879:
4878:
4873:
4871:
4870:
4869:
4868:
4867:
4866:
4803:
4801:
4800:
4795:
4793:
4792:
4791:
4790:
4776:
4775:
4737:
4735:
4734:
4729:
4727:
4726:
4725:
4724:
4711:
4710:
4709:
4684:
4682:
4681:
4676:
4665:
4664:
4663:
4642:
4640:
4639:
4634:
4629:
4628:
4605:
4603:
4602:
4597:
4595:
4594:
4570:
4568:
4567:
4562:
4551:
4550:
4549:
4521:
4519:
4518:
4513:
4511:
4510:
4509:
4508:
4494:
4493:
4467:
4465:
4464:
4459:
4457:
4456:
4455:
4454:
4444:
4443:
4442:
4420:
4418:
4417:
4412:
4400:
4395:
4368:
4367:
4319:
4317:
4316:
4311:
4299:
4294:
4258:
4256:
4255:
4250:
4239:
4238:
4218:
4216:
4215:
4210:
4199:
4198:
4147:
4145:
4144:
4139:
4128:
4127:
4104:
4102:
4101:
4096:
4085:
4084:
4063:
4062:
4040:
4038:
4037:
4032:
4021:
4020:
3942:
3940:
3939:
3934:
3929:
3928:
3924:
3923:
3922:
3921:
3847:
3845:
3844:
3839:
3834:
3833:
3815:
3814:
3794:
3792:
3791:
3786:
3775:
3774:
3750:
3748:
3747:
3742:
3737:
3736:
3698:
3696:
3695:
3690:
3685:
3684:
3680:
3679:
3678:
3677:
3639:
3637:
3636:
3631:
3629:
3628:
3602:
3600:
3599:
3594:
3592:
3591:
3564:
3562:
3561:
3556:
3527:
3526:
3495:
3493:
3492:
3487:
3415:
3413:
3412:
3407:
3402:
3401:
3377:
3376:
3338:
3336:
3335:
3330:
3328:
3327:
3326:
3325:
3305:
3284:
3282:
3281:
3276:
3274:
3273:
3272:
3249:
3247:
3246:
3241:
3239:
3238:
3237:
3220:
3218:
3217:
3212:
3210:
3209:
3208:
3187:
3185:
3184:
3179:
3177:
3176:
3175:
3174:
3160:
3159:
3130:= ( 10 β 3 β 3 )
3129:
3127:
3126:
3121:
3116:
3115:
3106:
3105:
3084:
3083:
3061:
3059:
3058:
3053:
3045:
3037:
3015:
3013:
3012:
3007:
3002:
3001:
2992:
2991:
2966:
2964:
2963:
2958:
2953:
2952:
2943:
2942:
2884:
2882:
2881:
2876:
2871:
2870:
2837:
2835:
2834:
2829:
2827:
2826:
2807:
2805:
2804:
2799:
2775:
2773:
2772:
2767:
2743:
2741:
2740:
2735:
2730:
2729:
2711:
2710:
2692:
2691:
2657:
2655:
2654:
2649:
2644:
2643:
2612:
2610:
2609:
2604:
2582:
2580:
2579:
2574:
2550:
2548:
2547:
2542:
2537:
2536:
2518:
2517:
2492:
2490:
2489:
2484:
2463:
2461:
2460:
2455:
2443:
2441:
2440:
2435:
2414:
2412:
2411:
2406:
2381:
2379:
2378:
2373:
2355:
2353:
2352:
2347:
2329:
2327:
2326:
2321:
2316:
2315:
2297:
2296:
2278:
2277:
2253:
2252:
2234:
2233:
2194:
2192:
2191:
2186:
2168:
2166:
2165:
2160:
2142:
2140:
2139:
2134:
2132:
2131:
2130:
2129:
2128:
2127:
2126:
2125:
2074:
2072:
2071:
2066:
2061:
2060:
2005:
2003:
2002:
1997:
1992:
1991:
1959:
1957:
1956:
1951:
1949:
1948:
1939:
1938:
1917:
1916:
1912:
1911:
1875:
1873:
1872:
1867:
1837:
1835:
1834:
1829:
1811:
1809:
1808:
1803:
1801:
1800:
1770:
1768:
1767:
1762:
1738:
1736:
1735:
1730:
1706:
1704:
1703:
1698:
1693:
1692:
1662:
1661:
1637:
1636:
1609:
1602:
1598:
1584:
1581:
1578:
1571:
1565:
1564:
1563:
1562:
1561:
1560:
1559:
1558:
1557:
1556:
1555:
1532:
1523:
1487:
1485:
1484:
1479:
1474:
1473:
1449:
1448:
1427:
1426:
1425:
1424:
1396:
1394:
1393:
1388:
1364:
1362:
1361:
1356:
1327:
1326:
1289:
1287:
1286:
1281:
1263:
1261:
1260:
1255:
1237:
1235:
1234:
1229:
1224:
1223:
1202:
1201:
1200:
1199:
1198:
1197:
1196:
1195:
1194:
1193:
1157:
1155:
1154:
1149:
1144:
1143:
1104:
1102:
1101:
1096:
1094:
1093:
1063:of the function
1061:functional power
1058:
1056:
1055:
1050:
1048:
1047:
1022:
1020:
1019:
1014:
1009:
1008:
940:
933:
929:
926:
920:
889:
881:
842:
840:
839:
834:
832:
831:
830:
829:
828:
827:
801:
799:
798:
793:
791:
790:
789:
788:
787:
786:
759:Skewes's numbers
755:
753:
752:
747:
745:
744:
743:
742:
725:
724:
721:
703:
701:
700:
695:
687:
686:
651:
647:
636:
634:
633:
628:
626:
625:
609:
607:
606:
601:
599:
598:
576:
574:
573:
568:
566:
565:
549:
547:
546:
541:
539:
538:
517:
515:
514:
509:
507:
506:
347:
345:
344:
339:
337:
333:
330:
329:
328:
327:
326:
325:
324:
323:
322:
321:
199:
197:
193:
169:
167:
124:
108:
96:
21:
12398:
12397:
12393:
12392:
12391:
12389:
12388:
12387:
12368:
12367:
12366:
12361:
12325:
12306:Subtraction (1)
12301:Predecessor (0)
12289:
12270:Subtraction (1)
12265:Predecessor (0)
12250:
12204:
12202:Hyperoperations
12199:
12169:
12164:
12146:
12102:List of numbers
12070:
12068:
12066:
12064:
12054:
12011:
11977:
11971:
11942:Graham's number
11932:Skewes's number
11834:
11832:
11830:
11824:
11819:
11789:
11788:
11779:
11777:
11768:
11767:
11763:
11754:
11752:
11744:
11743:
11739:
11730:
11728:
11720:
11719:
11715:
11686:
11682:
11667:
11663:
11658:
11655:
11654:
11652:
11648:
11635:
11634:
11630:
11625:Wayback Machine
11616:
11612:
11605:
11601:
11576:
11572:
11564:
11560:
11551:
11549:
11545:
11530:
11524:Shannon, Claude
11521:
11517:
11508:
11506:
11485:
11481:
11448:(6): e1002168.
11434:
11430:
11369:
11365:
11358:
11354:
11347:
11343:
11338:
11333:
11254:
11228:
11224:
11215:
11214:
11212:
11209:
11208:
11187:
11186:
11184:
11181:
11180:
11173:natural numbers
11159:. For example,
11145:
11131:
11129:cardinal number
11125:
11110:
11091:
11053:
11039:
11002:
10970:
10941:
10901:
10887:
10850:
10838:
10809:
10788:
10777:
10765:-1 β 1) (using
10740:
10729:
10719:
10710:
10695:
10685:
10675:
10668:
10660:Graham's number
10657:
10650:
10641:
10598:
10585:
10560:
10532:
10521:
10496:
10485:
10461:
10450:
10432:
10399:
10395:
10393:
10390:
10389:
10363:
10359:
10341:
10337:
10335:
10332:
10331:
10305:
10294:
10272:
10268:
10266:
10263:
10262:
10225:
10220:
10198:
10194:
10192:
10189:
10188:
10158:
10154:
10152:
10149:
10148:
10145:
10139:
10095:
10092:
10091:
10074:
10070:
10061:
10057:
10055:
10052:
10051:
10030:
10026:
10017:
10013:
10011:
10008:
10007:
9991:
9988:
9987:
9953:
9949:
9942:
9938:
9937:
9933:
9922:
9918:
9914:
9910:
9899:
9895:
9892:
9888:
9882:
9878:
9873:
9870:
9869:
9860:
9853:
9849:
9845:
9841:
9837:
9833:
9829:
9811:
9768:
9764:
9763:
9759:
9747:
9743:
9731:
9727:
9725:
9722:
9721:
9690:
9686:
9685:
9681:
9669:
9665:
9653:
9649:
9647:
9644:
9643:
9617:
9613:
9612:
9608:
9606:
9603:
9602:
9583:
9579:
9578:
9574:
9572:
9569:
9568:
9543:
9539:
9538:
9534:
9532:
9529:
9528:
9509:
9505:
9504:
9500:
9498:
9495:
9494:
9471:
9467:
9465:
9462:
9461:
9444:
9440:
9438:
9435:
9434:
9382:
9365:
9361:
9355:
9351:
9349:
9341:
9338:
9337:
9314:
9310:
9308:
9305:
9304:
9287:
9283:
9281:
9278:
9277:
9267:
9244:
9240:
9238:
9235:
9234:
9215:
9211:
9202:
9198:
9196:
9193:
9192:
9171:
9167:
9165:
9162:
9161:
9158:
9128:
9124:
9109:
9105:
9084:
9080:
9057:
9054:
9053:
9033:
9029:
9014:
9010:
8989:
8985:
8962:
8959:
8958:
8936:
8932:
8917:
8913:
8884:
8881:
8880:
8860:
8856:
8841:
8837:
8808:
8805:
8804:
8782:
8778:
8763:
8759:
8736:
8733:
8732:
8712:
8708:
8693:
8689:
8666:
8663:
8662:
8638:
8635:
8634:
8612:
8609:
8608:
8560:
8556:
8535:
8531:
8508:
8505:
8504:
8460:
8456:
8435:
8431:
8410:
8406:
8391:
8387:
8366:
8362:
8339:
8336:
8335:
8314:
8310:
8283:
8279:
8264:
8260:
8231:
8228:
8227:
8206:
8202:
8181:
8177:
8162:
8158:
8135:
8132:
8131:
8107:
8104:
8103:
8081:
8078:
8077:
8029:
8025:
7998:
7994:
7973:
7969:
7948:
7944:
7921:
7918:
7917:
7896:
7892:
7871:
7867:
7846:
7842:
7827:
7823:
7796:
7792:
7777:
7773:
7750:
7747:
7746:
7726:
7722:
7715:
7711:
7697:
7694:
7693:
7675:
7671:
7657:
7654:
7653:
7648:
7619:
7615:
7608:
7604:
7603:
7599:
7586:
7582:
7581:
7577:
7576:
7572:
7570:
7567:
7566:
7545:
7541:
7534:
7530:
7519:
7515:
7514:
7510:
7508:
7505:
7504:
7485:
7481:
7472:
7468:
7466:
7463:
7462:
7457:
7405:
7403:Other notations
7377:
7364:
7357:
7345:
7338:
7326:
7319:
7307:
7300:
7293:
7284:
7277:
7270:
7237:
7233:
7232:
7228:
7224:
7220:
7215:
7212:
7211:
7194:
7190:
7188:
7185:
7184:
7179:
7150:
7146:
7145:
7141:
7136:
7133:
7132:
7115:
7111:
7109:
7106:
7105:
7070:
7066:
7031:
7028:
7027:
7002:
6998:
6975:
6972:
6971:
6946:
6942:
6919:
6916:
6915:
6890:
6886:
6863:
6860:
6859:
6837:
6833:
6810:
6807:
6806:
6784:
6780:
6757:
6754:
6753:
6731:
6727:
6704:
6701:
6700:
6679:
6675:
6660:
6656:
6642:
6639:
6638:
6616:
6613:
6612:
6590:
6586:
6563:
6560:
6559:
6551:
6523:
6519:
6484:
6481:
6480:
6458:
6454:
6431:
6428:
6427:
6405:
6401:
6378:
6375:
6374:
6352:
6348:
6325:
6322:
6321:
6299:
6295:
6272:
6269:
6268:
6246:
6242:
6219:
6216:
6215:
6194:
6190:
6175:
6171:
6156:
6152:
6141:
6138:
6137:
6115:
6111:
6088:
6085:
6084:
6060:
6056:
6049:
6045:
6044:
6040:
6037:
6033:
6027:
6023:
6012:
6009:
6008:
5987:
5983:
5972:
5969:
5968:
5946:
5942:
5919:
5916:
5915:
5868:
5864:
5824:
5820:
5819:
5815:
5814:
5810:
5809:
5805:
5804:
5800:
5799:
5795:
5793:
5790:
5789:
5768:
5764:
5742:
5738:
5731:
5727:
5726:
5722:
5695:
5691:
5690:
5686:
5685:
5681:
5680:
5676:
5674:
5671:
5670:
5649:
5645:
5611:
5607:
5606:
5602:
5601:
5597:
5595:
5592:
5591:
5570:
5566:
5543:
5539:
5529:
5525:
5511:
5507:
5497:
5493:
5476:
5472:
5471:
5467:
5438:
5434:
5433:
5429:
5428:
5424:
5423:
5419:
5418:
5414:
5412:
5409:
5408:
5387:
5383:
5363:
5359:
5358:
5354:
5352:
5349:
5348:
5317:
5313:
5271:
5267:
5236:
5232:
5231:
5227:
5226:
5222:
5220:
5217:
5216:
5181:
5177:
5157:
5153:
5152:
5148:
5127:
5123:
5096:
5092:
5090:
5087:
5086:
5056:
5052:
5028:
5024:
5003:
4999:
4972:
4968:
4967:
4963:
4962:
4958:
4957:
4953:
4951:
4948:
4947:
4894:9 = 387,420,489
4862:
4858:
4857:
4853:
4852:
4848:
4846:
4843:
4842:
4827:2 = 2 ββ 3 = 16
4818:
4786:
4782:
4781:
4777:
4771:
4767:
4759:
4756:
4755:
4717:
4713:
4712:
4702:
4698:
4697:
4696:
4694:
4691:
4690:
4659:
4655:
4654:
4652:
4649:
4648:
4624:
4620:
4615:
4612:
4611:
4590:
4586:
4584:
4581:
4580:
4545:
4541:
4540:
4538:
4535:
4534:
4504:
4500:
4499:
4495:
4489:
4485:
4477:
4474:
4473:
4450:
4446:
4445:
4438:
4434:
4433:
4432:
4430:
4427:
4426:
4396:
4385:
4363:
4359:
4357:
4354:
4353:
4295:
4290:
4284:
4281:
4280:
4234:
4230:
4228:
4225:
4224:
4194:
4190:
4173:
4170:
4169:
4123:
4119:
4117:
4114:
4113:
4080:
4076:
4058:
4054:
4046:
4043:
4042:
4016:
4012:
3950:
3947:
3946:
3917:
3913:
3912:
3908:
3904:
3900:
3865:
3862:
3861:
3829:
3825:
3810:
3806:
3804:
3801:
3800:
3770:
3766:
3764:
3761:
3760:
3732:
3728:
3708:
3705:
3704:
3673:
3669:
3662:
3658:
3654:
3650:
3645:
3642:
3641:
3624:
3620:
3612:
3609:
3608:
3587:
3583:
3581:
3578:
3577:
3522:
3518:
3513:
3510:
3509:
3421:
3418:
3417:
3397:
3393:
3372:
3368:
3351:
3348:
3347:
3315:
3311:
3310:
3306:
3295:
3290:
3287:
3286:
3262:
3258:
3257:
3255:
3252:
3251:
3233:
3229:
3228:
3226:
3223:
3222:
3204:
3200:
3199:
3197:
3194:
3193:
3170:
3166:
3165:
3161:
3155:
3151:
3143:
3140:
3139:
3111:
3107:
3101:
3097:
3079:
3075:
3070:
3067:
3066:
3038:
3031:
3029:
3026:
3025:
2997:
2993:
2987:
2983:
2975:
2972:
2971:
2967:. For example:
2948:
2944:
2938:
2934:
2926:
2923:
2922:
2866:
2862:
2857:
2854:
2853:
2822:
2818:
2816:
2813:
2812:
2781:
2778:
2777:
2749:
2746:
2745:
2725:
2721:
2706:
2702:
2687:
2683:
2666:
2663:
2662:
2639:
2635:
2618:
2615:
2614:
2589:
2586:
2585:
2559:
2556:
2555:
2532:
2528:
2513:
2509:
2498:
2495:
2494:
2469:
2466:
2465:
2449:
2446:
2445:
2420:
2417:
2416:
2391:
2388:
2387:
2361:
2358:
2357:
2335:
2332:
2331:
2311:
2307:
2292:
2288:
2273:
2269:
2248:
2244:
2229:
2225:
2202:
2199:
2198:
2174:
2171:
2170:
2148:
2145:
2144:
2121:
2117:
2110:
2106:
2105:
2101:
2098:
2094:
2086:
2083:
2082:
2056:
2052:
2035:
2032:
2031:
2024:
2013:
1987:
1983:
1972:
1969:
1968:
1944:
1940:
1934:
1930:
1907:
1903:
1893:
1889:
1887:
1884:
1883:
1843:
1840:
1839:
1817:
1814:
1813:
1796:
1792:
1784:
1781:
1780:
1744:
1741:
1740:
1712:
1709:
1708:
1682:
1678:
1651:
1647:
1626:
1622:
1607:
1606:
1596:
1573:
1572:
1551:
1549:
1545:
1543:
1539:
1537:
1533:
1531:
1530:
1529:
1525:
1524:
1522:
1516:
1512:
1498:
1495:
1494:
1469:
1465:
1444:
1440:
1420:
1416:
1415:
1411:
1409:
1406:
1405:
1370:
1367:
1366:
1322:
1318:
1295:
1292:
1291:
1269:
1266:
1265:
1243:
1240:
1239:
1219:
1215:
1189:
1185:
1184:
1180:
1179:
1175:
1174:
1170:
1169:
1165:
1163:
1160:
1159:
1139:
1135:
1124:
1121:
1120:
1089:
1085:
1068:
1065:
1064:
1043:
1039:
1028:
1025:
1024:
1004:
1000:
959:
956:
955:
941:
930:
924:
921:
906:
890:
879:
860:Graham's number
846:Graham's number
823:
819:
818:
814:
813:
809:
807:
804:
803:
782:
778:
777:
773:
772:
768:
766:
763:
762:
738:
734:
733:
729:
720:
716:
714:
711:
710:
670:
666:
664:
661:
660:
649:
645:
621:
617:
615:
612:
611:
594:
590:
588:
585:
584:
561:
557:
555:
552:
551:
534:
530:
528:
525:
524:
502:
498:
496:
493:
492:
485:
441:
425:Harvey Friedman
331:
317:
313:
312:
308:
307:
303:
302:
298:
297:
293:
291:
288:
287:
248:
195:
191:
189:
165:
163:
122:
106:
94:
88:
70:
35:
28:
23:
22:
15:
12:
11:
5:
12396:
12386:
12385:
12380:
12363:
12362:
12360:
12359:
12354:
12349:
12344:
12339:
12333:
12331:
12327:
12326:
12324:
12323:
12318:
12313:
12308:
12303:
12297:
12295:
12291:
12290:
12288:
12287:
12285:Super-root (4)
12282:
12277:
12272:
12267:
12261:
12259:
12252:
12251:
12249:
12248:
12243:
12238:
12233:
12228:
12223:
12218:
12212:
12210:
12206:
12205:
12198:
12197:
12190:
12183:
12175:
12166:
12165:
12163:
12162:
12157:
12151:
12148:
12147:
12145:
12144:
12139:
12134:
12132:Power of three
12129:
12124:
12119:
12114:
12112:Number systems
12109:
12104:
12099:
12094:
12089:
12084:
12079:
12073:
12071:
12067:(alphabetical
12060:
12059:
12056:
12055:
12053:
12052:
12047:
12042:
12037:
12036:
12035:
12030:
12023:Hyperoperation
12019:
12017:
12013:
12012:
12010:
12009:
12004:
11999:
11994:
11988:
11986:
11979:
11973:
11972:
11970:
11969:
11964:
11959:
11954:
11949:
11944:
11939:
11937:Moser's number
11934:
11929:
11924:
11922:Shannon number
11919:
11914:
11909:
11904:
11899:
11894:
11889:
11884:
11879:
11874:
11869:
11864:
11859:
11854:
11849:
11844:
11838:
11836:
11826:
11825:
11818:
11817:
11810:
11803:
11795:
11787:
11786:
11761:
11737:
11713:
11700:
11695:
11692:
11689:
11685:
11681:
11678:
11675:
11670:
11666:
11662:
11646:
11628:
11610:
11599:
11587:hep-th/9411193
11570:
11558:
11526:(March 1950).
11515:
11479:
11428:
11383:(6): 463β471.
11363:
11352:
11340:
11339:
11337:
11334:
11332:
11331:
11326:
11321:
11316:
11311:
11306:
11301:
11296:
11294:Largest number
11291:
11286:
11281:
11276:
11271:
11266:
11261:
11255:
11253:
11250:
11231:
11227:
11223:
11218:
11190:
11139:Mahlo cardinal
11135:large cardinal
11127:Main article:
11124:
11121:
11090:
11087:
11084:
11083:
11048:
11037:
11032:
10997:
10992:
10968:
10963:
10936:
10931:
10896:
10885:
10880:
10845:
10840:
10834:
10805:
10786:
10775:
10770:
10738:
10733:
10732:
10731:
10724:
10715:
10708:
10693:
10680:
10673:
10666:
10655:
10648:
10643:
10639:
10594:
10583:
10578:
10555:
10550:
10530:
10519:
10514:
10494:
10483:
10478:
10459:
10448:
10430:
10421:ordinal number
10402:
10398:
10377:
10374:
10371:
10366:
10362:
10358:
10355:
10352:
10349:
10344:
10340:
10319:
10316:
10313:
10308:
10303:
10300:
10297:
10293:
10289:
10286:
10283:
10280:
10275:
10271:
10239:
10236:
10233:
10228:
10223:
10219:
10215:
10212:
10209:
10206:
10201:
10197:
10172:
10169:
10166:
10161:
10157:
10141:Main article:
10138:
10135:
10134:
10133:
10117:
10114:
10111:
10108:
10105:
10102:
10099:
10077:
10073:
10069:
10064:
10060:
10033:
10029:
10025:
10020:
10016:
9995:
9980:
9979:
9964:
9961:
9956:
9952:
9948:
9945:
9941:
9936:
9932:
9925:
9921:
9917:
9913:
9909:
9902:
9898:
9891:
9885:
9881:
9877:
9867:
9859:
9856:
9851:
9847:
9843:
9839:
9835:
9831:
9827:
9810:
9807:
9799:
9798:
9787:
9784:
9781:
9778:
9771:
9767:
9762:
9758:
9755:
9750:
9746:
9742:
9739:
9734:
9730:
9709:
9706:
9703:
9700:
9693:
9689:
9684:
9680:
9677:
9672:
9668:
9664:
9661:
9656:
9652:
9636:
9635:
9620:
9616:
9611:
9586:
9582:
9577:
9562:
9561:
9546:
9542:
9537:
9512:
9508:
9503:
9488:
9487:
9474:
9470:
9447:
9443:
9412:
9411:
9400:
9397:
9394:
9389:
9386:
9381:
9378:
9375:
9368:
9364:
9358:
9354:
9348:
9345:
9331:
9330:
9317:
9313:
9290:
9286:
9271:relative error
9266:
9263:
9247:
9243:
9218:
9214:
9210:
9207:
9201:
9174:
9170:
9157:
9154:
9153:
9152:
9136:
9131:
9127:
9123:
9120:
9117:
9112:
9108:
9104:
9101:
9098:
9093:
9090:
9087:
9083:
9079:
9076:
9073:
9070:
9067:
9064:
9061:
9041:
9036:
9032:
9028:
9025:
9022:
9017:
9013:
9009:
9006:
9003:
8998:
8995:
8992:
8988:
8984:
8981:
8978:
8975:
8972:
8969:
8966:
8956:
8944:
8939:
8935:
8931:
8928:
8925:
8920:
8916:
8912:
8909:
8906:
8903:
8900:
8897:
8894:
8891:
8888:
8868:
8863:
8859:
8855:
8852:
8849:
8844:
8840:
8836:
8833:
8830:
8827:
8824:
8821:
8818:
8815:
8812:
8802:
8790:
8785:
8781:
8777:
8774:
8771:
8766:
8762:
8758:
8755:
8752:
8749:
8746:
8743:
8740:
8720:
8715:
8711:
8707:
8704:
8701:
8696:
8692:
8688:
8685:
8682:
8679:
8676:
8673:
8670:
8660:
8648:
8645:
8642:
8622:
8619:
8616:
8592:
8589:
8586:
8583:
8580:
8577:
8574:
8571:
8568:
8563:
8559:
8555:
8552:
8549:
8544:
8541:
8538:
8534:
8530:
8527:
8524:
8521:
8518:
8515:
8512:
8492:
8489:
8486:
8483:
8480:
8477:
8474:
8471:
8468:
8463:
8459:
8455:
8452:
8449:
8444:
8441:
8438:
8434:
8430:
8427:
8424:
8421:
8418:
8413:
8409:
8405:
8402:
8399:
8394:
8390:
8386:
8383:
8380:
8375:
8372:
8369:
8365:
8361:
8358:
8355:
8352:
8349:
8346:
8343:
8333:
8322:
8317:
8313:
8309:
8306:
8303:
8300:
8297:
8294:
8291:
8286:
8282:
8278:
8275:
8272:
8267:
8263:
8259:
8256:
8253:
8250:
8247:
8244:
8241:
8238:
8235:
8225:
8214:
8209:
8205:
8201:
8198:
8195:
8192:
8189:
8184:
8180:
8176:
8173:
8170:
8165:
8161:
8157:
8154:
8151:
8148:
8145:
8142:
8139:
8129:
8117:
8114:
8111:
8091:
8088:
8085:
8061:
8058:
8055:
8052:
8049:
8046:
8043:
8040:
8037:
8032:
8028:
8024:
8021:
8018:
8015:
8012:
8007:
8004:
8001:
7997:
7993:
7990:
7987:
7984:
7981:
7976:
7972:
7968:
7965:
7962:
7957:
7954:
7951:
7947:
7943:
7940:
7937:
7934:
7931:
7928:
7925:
7915:
7904:
7899:
7895:
7891:
7888:
7885:
7882:
7879:
7874:
7870:
7866:
7863:
7860:
7857:
7854:
7849:
7845:
7841:
7838:
7835:
7830:
7826:
7822:
7819:
7816:
7813:
7810:
7807:
7804:
7799:
7795:
7791:
7788:
7785:
7780:
7776:
7772:
7769:
7766:
7763:
7760:
7757:
7754:
7744:
7729:
7725:
7721:
7718:
7714:
7710:
7707:
7704:
7701:
7691:
7678:
7674:
7670:
7667:
7664:
7661:
7646:
7630:
7627:
7622:
7618:
7614:
7611:
7607:
7602:
7598:
7589:
7585:
7580:
7575:
7548:
7544:
7540:
7537:
7533:
7529:
7522:
7518:
7513:
7488:
7484:
7480:
7475:
7471:
7456:
7453:
7444:
7443:
7438:
7428:
7423:
7416:hyperoperators
7404:
7401:
7400:
7399:
7396:
7393:
7390:
7387:
7384:
7381:
7378:
7375:
7369:
7366:
7362:
7355:
7350:
7347:
7343:
7336:
7331:
7328:
7324:
7317:
7312:
7309:
7305:
7298:
7291:
7286:
7282:
7275:
7268:
7263:
7252:
7247:
7240:
7236:
7231:
7227:
7223:
7219:
7197:
7193:
7181:
7177:
7171:
7160:
7153:
7149:
7144:
7140:
7118:
7114:
7102:
7099:
7096:
7093:
7090:
7078:
7073:
7069:
7065:
7062:
7059:
7056:
7053:
7050:
7047:
7044:
7041:
7038:
7035:
7025:
7013:
7010:
7005:
7001:
6997:
6994:
6991:
6988:
6985:
6982:
6979:
6969:
6957:
6954:
6949:
6945:
6941:
6938:
6935:
6932:
6929:
6926:
6923:
6913:
6901:
6898:
6893:
6889:
6885:
6882:
6879:
6876:
6873:
6870:
6867:
6857:
6845:
6840:
6836:
6832:
6829:
6826:
6823:
6820:
6817:
6814:
6804:
6792:
6787:
6783:
6779:
6776:
6773:
6770:
6767:
6764:
6761:
6751:
6739:
6734:
6730:
6726:
6723:
6720:
6717:
6714:
6711:
6708:
6698:
6687:
6682:
6678:
6674:
6671:
6668:
6663:
6659:
6655:
6652:
6649:
6646:
6626:
6623:
6620:
6610:
6609:= (10 β 3 β 4)
6598:
6593:
6589:
6585:
6582:
6579:
6576:
6573:
6570:
6567:
6557:
6549:
6543:
6531:
6526:
6522:
6518:
6515:
6512:
6509:
6506:
6503:
6500:
6497:
6494:
6491:
6488:
6478:
6466:
6461:
6457:
6453:
6450:
6447:
6444:
6441:
6438:
6435:
6425:
6413:
6408:
6404:
6400:
6397:
6394:
6391:
6388:
6385:
6382:
6372:
6360:
6355:
6351:
6347:
6344:
6341:
6338:
6335:
6332:
6329:
6319:
6307:
6302:
6298:
6294:
6291:
6288:
6285:
6282:
6279:
6276:
6266:
6254:
6249:
6245:
6241:
6238:
6235:
6232:
6229:
6226:
6223:
6213:
6202:
6197:
6193:
6189:
6186:
6183:
6178:
6174:
6170:
6167:
6164:
6159:
6155:
6151:
6148:
6145:
6135:
6123:
6118:
6114:
6110:
6107:
6104:
6101:
6098:
6095:
6092:
6082:
6063:
6059:
6055:
6052:
6048:
6043:
6036:
6030:
6026:
6022:
6019:
6016:
6006:
5995:
5990:
5986:
5982:
5979:
5976:
5966:
5954:
5949:
5945:
5941:
5938:
5935:
5932:
5929:
5926:
5923:
5913:
5906:
5905:
5902:
5899:
5896:
5893:
5890:
5887:
5876:
5871:
5867:
5863:
5860:
5857:
5854:
5851:
5848:
5845:
5842:
5827:
5823:
5818:
5813:
5808:
5803:
5798:
5787:
5776:
5771:
5767:
5763:
5760:
5757:
5754:
5745:
5741:
5737:
5734:
5730:
5725:
5721:
5718:
5715:
5712:
5709:
5698:
5694:
5689:
5684:
5679:
5668:
5657:
5652:
5648:
5644:
5641:
5638:
5635:
5632:
5629:
5626:
5623:
5614:
5610:
5605:
5600:
5589:
5578:
5573:
5569:
5565:
5562:
5559:
5556:
5551:
5546:
5542:
5538:
5535:
5532:
5528:
5524:
5519:
5514:
5510:
5506:
5503:
5500:
5496:
5492:
5485:
5482:
5479:
5475:
5470:
5466:
5463:
5460:
5457:
5454:
5441:
5437:
5432:
5427:
5422:
5417:
5406:
5395:
5390:
5386:
5382:
5379:
5376:
5373:
5366:
5362:
5357:
5337:
5336:
5325:
5320:
5316:
5312:
5309:
5306:
5303:
5298:
5295:
5292:
5289:
5286:
5283:
5280:
5277:
5274:
5270:
5266:
5263:
5260:
5257:
5254:
5251:
5248:
5239:
5235:
5230:
5225:
5214:
5211:
5208:
5205:Mersenne prime
5192:
5184:
5180:
5176:
5173:
5170:
5167:
5160:
5156:
5151:
5147:
5142:
5139:
5136:
5133:
5130:
5126:
5122:
5119:
5116:
5111:
5108:
5105:
5102:
5099:
5095:
5084:
5081:
5078:
5075:
5064:
5059:
5055:
5051:
5048:
5045:
5042:
5037:
5034:
5031:
5027:
5023:
5020:
5017:
5012:
5009:
5006:
5002:
4998:
4995:
4992:
4989:
4986:
4975:
4971:
4966:
4961:
4956:
4945:
4942:
4935:Planck volumes
4931:
4928:
4925:
4914:
4913:
4910:
4907:
4904:
4901:
4898:
4895:
4892:
4891:8 = 16,777,216
4889:
4886:
4883:
4865:
4861:
4856:
4851:
4840:
4837:
4834:
4831:
4828:
4825:
4817:
4814:
4789:
4785:
4780:
4774:
4770:
4766:
4763:
4723:
4720:
4716:
4708:
4705:
4701:
4674:
4671:
4668:
4662:
4658:
4632:
4627:
4623:
4619:
4593:
4589:
4560:
4557:
4554:
4548:
4544:
4507:
4503:
4498:
4492:
4488:
4484:
4481:
4453:
4449:
4441:
4437:
4410:
4407:
4404:
4399:
4394:
4391:
4388:
4384:
4380:
4377:
4374:
4371:
4366:
4362:
4309:
4306:
4303:
4298:
4293:
4289:
4248:
4245:
4242:
4237:
4233:
4208:
4205:
4202:
4197:
4193:
4189:
4186:
4183:
4180:
4177:
4137:
4134:
4131:
4126:
4122:
4094:
4091:
4088:
4083:
4079:
4075:
4072:
4069:
4066:
4061:
4057:
4053:
4050:
4030:
4027:
4024:
4019:
4015:
4011:
4008:
4005:
4002:
3999:
3996:
3993:
3990:
3987:
3984:
3981:
3978:
3975:
3972:
3969:
3966:
3963:
3960:
3957:
3954:
3932:
3927:
3920:
3916:
3911:
3907:
3903:
3899:
3896:
3893:
3890:
3887:
3884:
3881:
3878:
3875:
3872:
3869:
3837:
3832:
3828:
3824:
3821:
3818:
3813:
3809:
3784:
3781:
3778:
3773:
3769:
3740:
3735:
3731:
3727:
3724:
3721:
3718:
3715:
3712:
3700:
3699:
3688:
3683:
3676:
3672:
3668:
3665:
3661:
3657:
3653:
3649:
3627:
3623:
3619:
3616:
3590:
3586:
3554:
3551:
3548:
3545:
3542:
3539:
3536:
3533:
3530:
3525:
3521:
3517:
3485:
3482:
3479:
3476:
3473:
3470:
3467:
3464:
3461:
3458:
3455:
3452:
3449:
3446:
3443:
3440:
3437:
3434:
3431:
3428:
3425:
3405:
3400:
3396:
3392:
3389:
3386:
3383:
3380:
3375:
3371:
3367:
3364:
3361:
3358:
3355:
3339:is rewritten.
3324:
3321:
3318:
3314:
3309:
3304:
3301:
3298:
3294:
3271:
3268:
3265:
3261:
3236:
3232:
3207:
3203:
3173:
3169:
3164:
3158:
3154:
3150:
3147:
3132:
3131:
3119:
3114:
3110:
3104:
3100:
3096:
3093:
3090:
3087:
3082:
3078:
3074:
3051:
3048:
3044:
3041:
3036:
3033:
3022:
3021:
3005:
3000:
2996:
2990:
2986:
2982:
2979:
2956:
2951:
2947:
2941:
2937:
2933:
2930:
2911:
2910:
2874:
2869:
2865:
2861:
2843:hyper operator
2825:
2821:
2797:
2794:
2791:
2788:
2785:
2765:
2762:
2759:
2756:
2753:
2733:
2728:
2724:
2720:
2717:
2714:
2709:
2705:
2701:
2698:
2695:
2690:
2686:
2682:
2679:
2676:
2673:
2670:
2647:
2642:
2638:
2634:
2631:
2628:
2625:
2622:
2602:
2599:
2596:
2593:
2572:
2569:
2566:
2563:
2540:
2535:
2531:
2527:
2524:
2521:
2516:
2512:
2508:
2505:
2502:
2482:
2479:
2476:
2473:
2453:
2433:
2430:
2427:
2424:
2404:
2401:
2398:
2395:
2384:
2383:
2371:
2368:
2365:
2345:
2342:
2339:
2319:
2314:
2310:
2306:
2303:
2300:
2295:
2291:
2287:
2284:
2281:
2276:
2272:
2268:
2265:
2262:
2259:
2256:
2251:
2247:
2243:
2240:
2237:
2232:
2228:
2224:
2221:
2218:
2215:
2212:
2209:
2206:
2196:
2184:
2181:
2178:
2158:
2155:
2152:
2124:
2120:
2116:
2113:
2109:
2104:
2097:
2093:
2090:
2064:
2059:
2055:
2051:
2048:
2045:
2042:
2039:
2022:
2011:
1995:
1990:
1986:
1982:
1979:
1976:
1961:
1960:
1947:
1943:
1937:
1933:
1929:
1926:
1923:
1920:
1915:
1910:
1906:
1902:
1899:
1896:
1892:
1865:
1862:
1859:
1856:
1853:
1850:
1847:
1827:
1824:
1821:
1799:
1795:
1791:
1788:
1773:
1772:
1760:
1757:
1754:
1751:
1748:
1728:
1725:
1722:
1719:
1716:
1696:
1691:
1688:
1685:
1681:
1677:
1674:
1671:
1668:
1665:
1660:
1657:
1654:
1650:
1646:
1643:
1640:
1635:
1632:
1629:
1625:
1621:
1618:
1615:
1612:
1605:
1595:
1592:
1589:
1575:
1574:
1569:
1554:
1548:
1542:
1536:
1528:
1521:
1515:
1514:
1511:
1508:
1505:
1502:
1477:
1472:
1468:
1464:
1461:
1458:
1455:
1452:
1447:
1443:
1439:
1436:
1433:
1430:
1423:
1419:
1414:
1386:
1383:
1380:
1377:
1374:
1354:
1351:
1348:
1345:
1342:
1339:
1336:
1333:
1330:
1325:
1321:
1317:
1314:
1311:
1308:
1305:
1302:
1299:
1279:
1276:
1273:
1253:
1250:
1247:
1227:
1222:
1218:
1214:
1211:
1208:
1205:
1192:
1188:
1183:
1178:
1173:
1168:
1147:
1142:
1138:
1134:
1131:
1128:
1092:
1088:
1084:
1081:
1078:
1075:
1072:
1046:
1042:
1038:
1035:
1032:
1012:
1007:
1003:
999:
996:
993:
990:
987:
984:
981:
978:
975:
972:
969:
966:
963:
943:
942:
893:
891:
884:
878:
875:
874:
873:
863:
853:
843:
826:
822:
817:
812:
785:
781:
776:
771:
756:
741:
737:
732:
728:
719:
704:
693:
690:
685:
682:
679:
676:
673:
669:
657:Mersenne prime
653:
638:
624:
620:
597:
593:
578:
564:
560:
537:
533:
518:
505:
501:
484:
481:
440:
437:
375:repeats itself
365:model with an
355:
354:
353:
352:
351:
350:
349:
348:
320:
316:
311:
306:
301:
296:
260:Big Bang model
247:
244:
243:
242:
241:
240:
237:Taikyoku Shogi
222:Shannon number
218:
207:
201:
171:
148:
143:The number of
141:
132:on a computer
128:The number of
126:
117:The number of
69:
66:
26:
9:
6:
4:
3:
2:
12395:
12384:
12381:
12379:
12378:Large numbers
12376:
12375:
12373:
12358:
12355:
12353:
12350:
12348:
12345:
12343:
12340:
12338:
12335:
12334:
12332:
12328:
12322:
12319:
12317:
12316:Logarithm (3)
12314:
12312:
12309:
12307:
12304:
12302:
12299:
12298:
12296:
12292:
12286:
12283:
12281:
12278:
12276:
12273:
12271:
12268:
12266:
12263:
12262:
12260:
12257:
12253:
12247:
12244:
12242:
12241:Pentation (5)
12239:
12237:
12236:Tetration (4)
12234:
12232:
12229:
12227:
12224:
12222:
12219:
12217:
12216:Successor (0)
12214:
12213:
12211:
12207:
12203:
12196:
12191:
12189:
12184:
12182:
12177:
12176:
12173:
12161:
12158:
12156:
12153:
12152:
12149:
12143:
12140:
12138:
12135:
12133:
12130:
12128:
12125:
12123:
12120:
12118:
12115:
12113:
12110:
12108:
12105:
12103:
12100:
12098:
12095:
12093:
12092:Infinitesimal
12090:
12088:
12085:
12083:
12080:
12078:
12075:
12074:
12072:
12061:
12051:
12048:
12046:
12043:
12041:
12038:
12034:
12031:
12029:
12026:
12025:
12024:
12021:
12020:
12018:
12014:
12008:
12005:
12003:
12000:
11998:
11995:
11993:
11990:
11989:
11987:
11983:
11980:
11974:
11968:
11965:
11963:
11962:Rayo's number
11960:
11958:
11955:
11953:
11950:
11948:
11945:
11943:
11940:
11938:
11935:
11933:
11930:
11928:
11925:
11923:
11920:
11918:
11915:
11913:
11910:
11908:
11905:
11903:
11900:
11898:
11895:
11893:
11890:
11888:
11885:
11883:
11880:
11878:
11875:
11873:
11870:
11868:
11865:
11863:
11860:
11858:
11855:
11853:
11850:
11848:
11845:
11843:
11840:
11839:
11837:
11827:
11823:
11822:Large numbers
11816:
11811:
11809:
11804:
11802:
11797:
11796:
11793:
11775:
11771:
11765:
11751:
11747:
11741:
11727:
11723:
11717:
11698:
11693:
11690:
11687:
11679:
11676:
11673:
11668:
11660:
11650:
11643:. 2018-12-21.
11642:
11638:
11632:
11626:
11622:
11619:
11614:
11608:
11603:
11596:
11595:0-9630728-3-8
11592:
11588:
11584:
11580:
11574:
11567:
11562:
11548:on 2010-07-06
11544:
11540:
11536:
11529:
11525:
11519:
11505:
11501:
11497:
11493:
11489:
11483:
11475:
11471:
11466:
11461:
11456:
11451:
11447:
11443:
11439:
11432:
11424:
11420:
11416:
11412:
11408:
11404:
11400:
11396:
11391:
11386:
11382:
11378:
11374:
11367:
11361:
11356:
11350:
11345:
11341:
11330:
11327:
11325:
11322:
11320:
11317:
11315:
11312:
11310:
11307:
11305:
11302:
11300:
11297:
11295:
11292:
11290:
11287:
11285:
11282:
11280:
11277:
11275:
11272:
11270:
11267:
11265:
11262:
11260:
11257:
11256:
11249:
11247:
11229:
11221:
11206:
11178:
11174:
11170:
11166:
11162:
11158:
11154:
11150:
11144:
11140:
11136:
11130:
11120:
11118:
11113:
11108:
11104:
11100:
11096:
11086:
11081:
11077:
11073:
11069:
11065:
11061:
11057:
11052:
11047:
11043:
11036:
11033:
11030:
11026:
11022:
11018:
11014:
11010:
11006:
11001:
10996:
10993:
10990:
10986:
10982:
10978:
10974:
10967:
10964:
10961:
10957:
10953:
10949:
10945:
10940:
10935:
10932:
10929:
10925:
10921:
10917:
10913:
10909:
10905:
10900:
10895:
10891:
10884:
10881:
10878:
10874:
10870:
10866:
10862:
10858:
10854:
10849:
10844:
10841:
10837:
10833:
10829:
10825:
10821:
10817:
10813:
10808:
10804:
10800:
10796:
10792:
10785:
10781:
10774:
10771:
10768:
10764:
10760:
10757:-1) = (2 β
10756:
10752:
10748:
10744:
10737:
10734:
10727:
10723:
10718:
10714:
10707:
10703:
10699:
10692:
10688:
10687:
10683:
10679:
10672:
10665:
10661:
10654:
10647:
10644:
10638:
10634:
10630:
10626:
10622:
10618:
10614:
10610:
10606:
10602:
10597:
10593:
10589:
10582:
10579:
10576:
10572:
10568:
10564:
10558:
10554:
10551:
10548:
10544:
10540:
10537:) > (2 β)
10536:
10529:
10525:
10518:
10515:
10512:
10508:
10504:
10500:
10493:
10489:
10482:
10479:
10477:
10473:
10469:
10465:
10458:
10454:
10447:
10444:
10443:
10442:
10440:
10436:
10429:
10424:
10422:
10418:
10400:
10396:
10372:
10364:
10360:
10356:
10350:
10342:
10338:
10314:
10306:
10301:
10298:
10295:
10291:
10287:
10281:
10273:
10269:
10260:
10257:
10253:
10234:
10226:
10221:
10217:
10213:
10207:
10199:
10195:
10186:
10167:
10159:
10155:
10144:
10131:
10115:
10109:
10106:
10103:
10097:
10075:
10071:
10067:
10062:
10058:
10049:
10031:
10027:
10023:
10018:
10014:
9993:
9985:
9984:
9983:
9962:
9959:
9954:
9950:
9946:
9943:
9939:
9934:
9930:
9923:
9919:
9915:
9911:
9907:
9900:
9896:
9883:
9879:
9868:
9865:
9864:
9863:
9855:
9823:
9821:
9817:
9806:
9804:
9785:
9782:
9769:
9765:
9760:
9753:
9748:
9744:
9737:
9732:
9728:
9707:
9704:
9691:
9687:
9682:
9675:
9670:
9666:
9659:
9654:
9650:
9642:
9641:
9640:
9618:
9614:
9609:
9584:
9580:
9575:
9567:
9566:
9565:
9544:
9540:
9535:
9510:
9506:
9501:
9493:
9492:
9491:
9472:
9468:
9445:
9441:
9433:
9432:
9431:
9429:
9425:
9421:
9416:
9395:
9392:
9387:
9384:
9379:
9376:
9373:
9366:
9362:
9356:
9352:
9346:
9343:
9336:
9335:
9334:
9315:
9311:
9288:
9284:
9276:
9275:
9274:
9272:
9262:
9245:
9241:
9216:
9212:
9208:
9205:
9199:
9190:
9172:
9168:
9160:For a number
9150:
9134:
9129:
9118:
9110:
9099:
9091:
9088:
9085:
9074:
9068:
9065:
9059:
9039:
9034:
9023:
9015:
9004:
8996:
8993:
8990:
8979:
8973:
8970:
8964:
8957:
8942:
8937:
8926:
8918:
8907:
8898:
8895:
8892:
8886:
8866:
8861:
8850:
8842:
8831:
8822:
8819:
8816:
8810:
8803:
8788:
8783:
8772:
8764:
8753:
8747:
8744:
8738:
8718:
8713:
8702:
8694:
8683:
8677:
8674:
8668:
8661:
8646:
8640:
8620:
8614:
8606:
8587:
8584:
8581:
8572:
8569:
8566:
8561:
8550:
8542:
8539:
8536:
8525:
8519:
8516:
8510:
8487:
8484:
8481:
8472:
8469:
8466:
8461:
8450:
8442:
8439:
8436:
8425:
8419:
8411:
8407:
8403:
8400:
8392:
8381:
8373:
8370:
8367:
8356:
8350:
8347:
8341:
8334:
8320:
8315:
8304:
8295:
8292:
8284:
8280:
8276:
8273:
8265:
8254:
8245:
8242:
8239:
8233:
8226:
8212:
8207:
8196:
8190:
8182:
8178:
8174:
8171:
8163:
8152:
8146:
8143:
8137:
8130:
8115:
8109:
8089:
8083:
8075:
8056:
8053:
8050:
8041:
8038:
8035:
8030:
8019:
8013:
8010:
8005:
8002:
7999:
7988:
7982:
7974:
7970:
7966:
7963:
7955:
7952:
7949:
7938:
7932:
7929:
7923:
7916:
7902:
7897:
7886:
7880:
7877:
7872:
7861:
7855:
7847:
7843:
7839:
7836:
7828:
7817:
7811:
7805:
7802:
7797:
7793:
7789:
7786:
7778:
7767:
7761:
7758:
7752:
7745:
7727:
7723:
7719:
7716:
7712:
7708:
7705:
7699:
7692:
7676:
7672:
7668:
7665:
7659:
7652:
7651:
7650:
7628:
7625:
7620:
7616:
7612:
7609:
7605:
7600:
7596:
7587:
7583:
7578:
7573:
7564:
7546:
7542:
7538:
7535:
7531:
7527:
7520:
7516:
7511:
7502:
7486:
7482:
7478:
7473:
7469:
7460:
7452:
7448:
7442:
7439:
7436:
7432:
7429:
7427:
7424:
7421:
7417:
7413:
7410:
7409:
7408:
7397:
7394:
7391:
7388:
7385:
7382:
7379:
7374:
7370:
7367:
7361:
7354:
7351:
7348:
7342:
7335:
7332:
7329:
7323:
7316:
7313:
7310:
7304:
7297:
7290:
7287:
7281:
7274:
7267:
7264:
7250:
7245:
7238:
7234:
7225:
7217:
7195:
7191:
7182:
7176:
7172:
7158:
7151:
7147:
7138:
7116:
7112:
7103:
7100:
7097:
7094:
7091:
7076:
7071:
7060:
7054:
7051:
7045:
7042:
7039:
7033:
7026:
7011:
7008:
7003:
6992:
6986:
6983:
6977:
6970:
6955:
6952:
6947:
6936:
6930:
6927:
6921:
6914:
6899:
6896:
6891:
6880:
6874:
6871:
6865:
6858:
6843:
6838:
6827:
6821:
6818:
6812:
6805:
6790:
6785:
6774:
6768:
6765:
6759:
6752:
6737:
6732:
6721:
6715:
6712:
6706:
6699:
6685:
6680:
6669:
6661:
6650:
6644:
6624:
6618:
6611:
6596:
6591:
6580:
6574:
6571:
6565:
6558:
6555:
6548:
6544:
6529:
6524:
6513:
6507:
6504:
6498:
6495:
6492:
6486:
6479:
6464:
6459:
6448:
6442:
6439:
6433:
6426:
6411:
6406:
6395:
6389:
6386:
6380:
6373:
6358:
6353:
6342:
6336:
6333:
6327:
6320:
6305:
6300:
6289:
6283:
6280:
6274:
6267:
6252:
6247:
6236:
6230:
6227:
6221:
6214:
6195:
6191:
6187:
6184:
6176:
6165:
6157:
6146:
6136:
6121:
6116:
6105:
6099:
6096:
6090:
6083:
6061:
6057:
6053:
6050:
6046:
6041:
6034:
6028:
6017:
6007:
5993:
5988:
5977:
5967:
5952:
5947:
5936:
5930:
5927:
5921:
5914:
5911:
5910:
5909:
5903:
5900:
5897:
5894:
5891:
5888:
5874:
5869:
5858:
5852:
5849:
5843:
5840:
5825:
5821:
5816:
5811:
5806:
5801:
5796:
5788:
5774:
5769:
5758:
5752:
5743:
5739:
5735:
5732:
5728:
5723:
5719:
5716:
5710:
5707:
5696:
5692:
5687:
5682:
5677:
5669:
5655:
5650:
5639:
5633:
5630:
5624:
5621:
5612:
5608:
5603:
5598:
5590:
5576:
5571:
5560:
5554:
5549:
5544:
5533:
5526:
5522:
5517:
5512:
5501:
5494:
5490:
5483:
5480:
5477:
5473:
5468:
5464:
5461:
5455:
5452:
5439:
5435:
5430:
5425:
5420:
5415:
5407:
5393:
5388:
5377:
5371:
5364:
5360:
5355:
5347:googolplex =
5346:
5345:
5344:
5342:
5323:
5318:
5307:
5301:
5296:
5293:
5290:
5287:
5284:
5281:
5278:
5275:
5272:
5268:
5264:
5261:
5258:
5255:
5249:
5246:
5237:
5233:
5228:
5223:
5215:
5212:
5209:
5206:
5190:
5182:
5171:
5165:
5158:
5154:
5149:
5145:
5140:
5137:
5134:
5131:
5128:
5124:
5120:
5117:
5114:
5109:
5106:
5103:
5100:
5097:
5093:
5085:
5082:
5079:
5076:
5062:
5057:
5046:
5040:
5035:
5032:
5029:
5025:
5021:
5018:
5015:
5010:
5007:
5004:
5000:
4996:
4993:
4987:
4984:
4973:
4969:
4964:
4959:
4954:
4946:
4943:
4940:
4936:
4932:
4929:
4926:
4923:
4919:
4918:
4917:
4911:
4908:
4905:
4902:
4899:
4896:
4893:
4890:
4887:
4884:
4863:
4859:
4854:
4849:
4841:
4838:
4835:
4832:
4829:
4826:
4823:
4822:
4821:
4813:
4809:
4807:
4787:
4783:
4772:
4764:
4753:
4749:
4745:
4741:
4721:
4718:
4714:
4706:
4703:
4699:
4688:
4669:
4660:
4656:
4646:
4630:
4625:
4617:
4609:
4591:
4587:
4578:
4574:
4555:
4546:
4542:
4532:
4527:
4525:
4505:
4501:
4490:
4482:
4471:
4451:
4447:
4439:
4435:
4424:
4405:
4397:
4392:
4389:
4386:
4382:
4378:
4372:
4364:
4360:
4351:
4347:
4343:
4339:
4335:
4331:
4327:
4323:
4304:
4296:
4291:
4287:
4278:
4274:
4270:
4266:
4262:
4243:
4235:
4231:
4222:
4203:
4195:
4191:
4187:
4181:
4175:
4167:
4163:
4159:
4155:
4151:
4132:
4124:
4120:
4111:
4106:
4089:
4081:
4077:
4073:
4067:
4059:
4055:
4051:
4048:
4025:
4017:
4013:
4009:
4003:
3997:
3991:
3985:
3979:
3976:
3970:
3964:
3958:
3955:
3952:
3943:
3930:
3925:
3918:
3914:
3905:
3897:
3894:
3888:
3882:
3876:
3870:
3859:
3856:(1) = (10β10β
3855:
3851:
3830:
3826:
3822:
3819:
3811:
3807:
3798:
3779:
3771:
3767:
3758:
3754:
3751:= (10 β 10 β
3738:
3733:
3725:
3722:
3716:
3710:
3686:
3681:
3674:
3670:
3666:
3663:
3655:
3647:
3625:
3621:
3617:
3614:
3606:
3605:
3604:
3588:
3575:
3570:
3568:
3549:
3543:
3537:
3531:
3528:
3523:
3515:
3505:
3503:
3499:
3483:
3477:
3474:
3468:
3465:
3462:
3456:
3447:
3444:
3438:
3432:
3423:
3403:
3398:
3387:
3381:
3378:
3373:
3362:
3353:
3345:
3340:
3322:
3319:
3316:
3312:
3302:
3299:
3296:
3269:
3266:
3263:
3259:
3234:
3230:
3205:
3201:
3191:
3171:
3167:
3156:
3148:
3137:
3117:
3112:
3102:
3094:
3088:
3085:
3080:
3072:
3065:
3064:
3063:
3049:
3046:
3042:
3039:
3034:
3032:
3019:
3003:
2998:
2988:
2980:
2970:
2969:
2968:
2954:
2949:
2939:
2931:
2920:
2916:
2908:
2904:
2900:
2896:
2892:
2888:
2872:
2867:
2859:
2852:
2851:
2850:
2848:
2844:
2839:
2823:
2809:
2795:
2789:
2783:
2763:
2757:
2751:
2726:
2722:
2718:
2715:
2707:
2696:
2688:
2677:
2668:
2659:
2640:
2636:
2632:
2629:
2620:
2594:
2564:
2552:
2533:
2529:
2525:
2522:
2514:
2503:
2474:
2451:
2425:
2396:
2369:
2363:
2343:
2337:
2312:
2308:
2304:
2301:
2293:
2282:
2274:
2263:
2257:
2249:
2245:
2241:
2238:
2230:
2219:
2210:
2204:
2197:
2182:
2176:
2156:
2150:
2122:
2118:
2114:
2111:
2107:
2102:
2095:
2088:
2081:
2080:
2079:
2076:
2057:
2053:
2049:
2046:
2037:
2027:
2020:
2016:
2009:
1993:
1988:
1977:
1966:
1945:
1941:
1935:
1924:
1918:
1913:
1908:
1897:
1890:
1882:
1881:
1880:
1877:
1860:
1857:
1854:
1845:
1825:
1819:
1797:
1793:
1789:
1786:
1778:
1758:
1755:
1752:
1746:
1726:
1723:
1720:
1714:
1694:
1689:
1686:
1683:
1672:
1666:
1658:
1655:
1652:
1648:
1644:
1641:
1633:
1630:
1627:
1616:
1610:
1603:
1593:
1590:
1587:
1567:
1552:
1546:
1540:
1534:
1526:
1519:
1509:
1506:
1500:
1493:
1492:
1491:
1488:
1475:
1470:
1459:
1453:
1450:
1445:
1434:
1428:
1421:
1417:
1412:
1402:
1400:
1397:). (See also
1384:
1381:
1378:
1375:
1372:
1349:
1346:
1343:
1334:
1331:
1328:
1323:
1312:
1306:
1303:
1297:
1277:
1271:
1251:
1245:
1225:
1220:
1209:
1203:
1190:
1186:
1181:
1176:
1171:
1166:
1145:
1140:
1129:
1117:
1115:
1110:
1108:
1090:
1086:
1082:
1076:
1070:
1062:
1044:
1033:
1010:
1005:
994:
988:
985:
979:
973:
970:
967:
961:
952:
948:
939:
936:
928:
925:December 2022
918:
914:
910:
904:
903:
899:
894:This section
892:
888:
883:
882:
871:
867:
866:Rayo's number
864:
861:
857:
854:
851:
847:
844:
824:
820:
815:
810:
802:, the second
783:
779:
774:
769:
760:
757:
739:
735:
730:
726:
717:
708:
705:
691:
688:
683:
680:
677:
674:
671:
667:
658:
654:
643:
639:
622:
618:
595:
591:
582:
579:
562:
558:
535:
531:
522:
519:
503:
499:
490:
487:
486:
480:
478:
474:
470:
466:
465:Johnny Carson
462:
461:
456:
452:
448:
447:
436:
434:
430:
426:
421:
419:
415:
411:
410:GΓΆdel numbers
407:
405:
400:
398:
394:
390:
386:
385:Combinatorial
382:
380:
376:
372:
371:Planck masses
368:
364:
360:
318:
314:
309:
304:
299:
294:
286:
285:
284:
283:
282:
281:
280:
279:
278:
276:
273:According to
271:
269:
265:
261:
257:
253:
238:
234:
230:
229:Grant Acedrex
226:
225:
223:
219:
216:
212:
208:
206:
202:
187:
183:
179:
176:
172:
161:
157:
153:
149:
146:
142:
139:
135:
131:
127:
120:
116:
115:
114:
111:
104:
100:
92:
87:
83:
79:
75:
65:
63:
59:
55:
51:
47:
43:
39:
38:Large numbers
33:
19:
12311:Division (2)
12275:Division (2)
12246:Hexation (6)
12221:Addition (1)
12127:Power of two
12117:Number names
11852:Ten thousand
11821:
11778:. Retrieved
11776:. 2024-07-02
11773:
11764:
11753:. Retrieved
11750:www.mrob.com
11749:
11740:
11729:. Retrieved
11726:www.mrob.com
11725:
11716:
11649:
11640:
11631:
11613:
11602:
11578:
11573:
11561:
11550:. Retrieved
11543:the original
11538:
11537:. Series 7.
11534:
11518:
11507:. Retrieved
11498:. New York.
11495:
11482:
11445:
11442:PLOS Biology
11441:
11431:
11399:11585/152451
11380:
11376:
11366:
11355:
11344:
11319:Power of two
11169:infinite set
11146:
11106:
11102:
11092:
11085:
11079:
11075:
11074:) (Chain of
11071:
11067:
11063:
11059:
11055:
11050:
11045:
11041:
11034:
11028:
11024:
11023:) (Chain of
11020:
11016:
11012:
11008:
11004:
10999:
10994:
10988:
10984:
10980:
10976:
10972:
10965:
10959:
10955:
10951:
10947:
10943:
10938:
10933:
10927:
10923:
10919:
10915:
10911:
10907:
10903:
10898:
10893:
10889:
10882:
10876:
10872:
10868:
10864:
10860:
10856:
10855:) > (2 β
10852:
10847:
10842:
10835:
10831:
10827:
10823:
10819:
10815:
10811:
10806:
10802:
10798:
10794:
10793:) > (2 β
10790:
10783:
10779:
10772:
10762:
10758:
10754:
10750:
10746:
10742:
10735:
10725:
10721:
10716:
10712:
10705:
10701:
10697:
10690:
10681:
10677:
10670:
10663:
10652:
10645:
10636:
10628:
10624:
10620:
10616:
10612:
10608:
10604:
10600:
10595:
10591:
10587:
10580:
10574:
10570:
10566:
10562:
10556:
10552:
10546:
10542:
10538:
10534:
10527:
10523:
10516:
10506:
10502:
10498:
10491:
10487:
10480:
10475:
10471:
10467:
10463:
10456:
10452:
10445:
10438:
10434:
10427:
10425:
10416:
10255:
10254:denotes the
10251:
10184:
10146:
9981:
9861:
9824:
9812:
9800:
9637:
9563:
9489:
9427:
9423:
9417:
9413:
9332:
9268:
9188:
9159:
9148:
8604:
8073:
7565:
7503:
7461:
7458:
7451:reciprocal.
7449:
7445:
7406:
7372:
7359:
7358:= (3 β 3 β
7352:
7340:
7339:= (3 β 3 β
7333:
7321:
7320:= (3 β 3 β
7314:
7302:
7295:
7294:= (3 β 3 β
7288:
7279:
7272:
7271:= (3 β 3 β
7265:
7174:
6546:
5907:
5340:
5338:
4920:Approximate
4915:
4819:
4810:
4805:
4751:
4747:
4743:
4738:, nested in
4686:
4644:
4607:
4576:
4572:
4530:
4528:
4523:
4469:
4422:
4349:
4345:
4341:
4337:
4333:
4329:
4325:
4321:
4276:
4272:
4268:
4264:
4260:
4220:
4165:
4161:
4157:
4153:
4149:
4109:
4107:
3944:
3857:
3853:
3849:
3796:
3756:
3752:
3701:
3573:
3571:
3566:
3506:
3501:
3497:
3343:
3341:
3189:
3135:
3133:
3023:
3017:
2918:
2914:
2912:
2906:
2902:
2898:
2894:
2890:
2886:
2840:
2810:
2660:
2553:
2385:
2077:
2028:
2018:
2014:
2007:
1964:
1962:
1878:
1776:
1774:
1489:
1403:
1118:
1111:
953:
949:
946:
931:
922:
907:Please help
895:
642:Smith number
581:millinillion
472:
460:Tonight Show
458:
444:
442:
422:
418:pathological
408:
401:
393:permutations
383:
363:inflationary
356:
272:
249:
246:Astronomical
186:biodiversity
112:
89:
61:
50:cryptography
37:
36:
18:Large number
12137:Power of 10
12077:Busy beaver
11882:Quintillion
11877:Quadrillion
11324:Power of 10
11284:Human scale
11165:cardinality
11095:busy beaver
10745:) > 2 β
10700:) > 2 β
10627:β₯ 2, where
10611:+ 3) β 3 =
10603:) > 2 β
10565:) > 2 β
10505:> (2 β)
10415:for finite
10050:) and also
9842:) and 2log(
9820:cardinality
4885:7 = 823,543
4685:=(10β10β10β
4571:=(10β10β10β
4332:=1 for the
4041:, but also
1290:(note that
334: years
268:light years
42:mathematics
12372:Categories
12142:Sagan Unit
11976:Expression
11927:Googolplex
11892:Septillion
11887:Sextillion
11833:numerical
11780:2024-07-04
11755:2021-05-13
11731:2021-05-13
11552:2019-01-25
11509:2015-07-18
11336:References
11161:aleph-null
11133:See also:
11099:computable
10867:+1) > (
10720:+ 2) >
10651:(64) >
10635:(of which
10607:> 2 β (
10046:(see e.g.
8603:; thus if
5343:notation:
4941:= 8.5 Γ 10
4839:6 = 46,656
3134:Since the
2897:) = hyper(
2078:Examples:
2017:, or find
1879:Note that
1059:denotes a
707:googolplex
521:centillion
477:Sagan Unit
451:Carl Sagan
404:logarithms
178:base pairs
103:reciprocal
72:See also:
12033:Pentation
12028:Tetration
12016:Operators
11985:Notations
11907:Decillion
11902:Nonillion
11897:Octillion
11829:Examples
11684:↑
11665:↑
11504:0362-4331
11407:1464-5033
11329:Tetration
11226:ℵ
11177:aleph-one
11058:) > (
10822:then X β
10686:= 3 β 3)
10658:(6) >
10343:ω
10299:−
10107:≈
10068:≈
10024:≈
10006:there is
9960:
9816:subitized
9754:
9738:
9676:
9660:
9399:%
9380:−
9347:−
9209:×
9122:↑
9089:−
9052:(compare
9027:↑
8994:−
8930:↑
8879:(compare
8854:↑
8776:↑
8731:(compare
8706:↑
8554:↑
8540:−
8503:(compare
8454:↑
8440:−
8404:×
8385:↑
8371:−
8308:↑
8277:×
8258:↑
8200:↑
8175:×
8156:↑
8072:(thus if
8023:↑
8003:−
7992:↑
7967:×
7953:−
7942:↑
7890:↑
7865:↑
7840:×
7821:↑
7790:×
7771:↑
7720:×
7613:∗
7597:≈
7563:, ditto.
7539:∗
7230:↑
7222:↑
7143:↑
6645:≈
6188:×
6169:↑
6054:×
5862:↑
5853:≈
5762:↑
5753:≈
5736:×
5720:≈
5643:↑
5564:↑
5537:↑
5523:≈
5505:↑
5491:≈
5381:↑
5311:↑
5302:≈
5265:×
5259:≈
5175:↑
5146:≈
5121:×
5115:≈
5050:↑
5041:≈
5022:×
5016:≈
4836:5 = 3,125
4769:↑
4622:↑
4487:↑
4390:−
4001:→
3995:→
3989:→
3974:→
3968:→
3962:→
3910:↑
3902:↑
3886:→
3880:→
3874:→
3823:×
3730:↑
3667:×
3660:↑
3652:↑
3618:×
3585:↑
3547:→
3541:→
3520:↑
3475:≈
3427:↑
3357:↑
3153:↑
3099:↑
3077:↑
3062:obtains:
2985:↑
2936:↑
2921:arrows):
2864:↑
2820:↑
2744:(between
2719:×
2700:↑
2633:×
2526:×
2429:↑
2400:↑
2330:(between
2305:×
2286:↑
2242:×
2223:↑
2143:(between
2115:×
2050:×
1981:↑
1928:↑
1901:↑
1707:(between
1676:↑
1667:≈
1645:×
1620:↑
1611:≈
1568:⏟
1463:↑
1438:↑
1316:↑
1213:↑
1133:↑
1037:↑
998:↑
983:→
977:→
896:does not
850:tetration
689:−
423:Logician
389:factorial
256:cosmology
252:astronomy
233:Tai Shogi
160:carbon-12
134:hard disk
62:Googology
46:cosmology
12065:articles
12063:Related
11967:Infinity
11872:Trillion
11847:Thousand
11621:Archived
11474:26066900
11423:16247166
11415:23829164
11299:Infinity
11252:See also
11153:infinite
11066:β ... β
11015:β ... β
11007:) > (
10975:) > (
10946:) > (
10906:) > (
10814:) = X β
10749:= (2 β
9156:Accuracy
7435:polygons
4939:universe
4816:Examples
4750:and for
4643:=(10β10β
4271:(1). If
3043:″
3035:″
2845:and the
648:10 + 1)
483:Examples
431:and the
367:inflaton
275:Don Page
205:nucleons
58:integers
12256:Inverse
12209:Primary
12160:History
11978:methods
11952:SSCG(3)
11947:TREE(3)
11867:Billion
11862:Million
11842:Hundred
11488:Nuwer R
11465:4466264
11203:is the
11167:of the
11163:is the
11115:in the
11112:A028444
10631:is the
10090:. Thus
9982:Hence:
9809:Classes
7629:0.30103
4833:4 = 256
4340:=2 for
4336:above,
4152:, e.g.
2901:,
917:removed
902:sources
213:in the
182:biomass
99:billion
12069:order)
11917:Googol
11593:
11502:
11472:
11462:
11421:
11413:
11405:
11175:, and
11149:finite
11141:, and
11105:) for
10623:) for
10577:< Ο
10541:β₯ 2 β
10128:, see
7210:) ) =
5191:7.3955
5188:
5159:7.3955
4830:3 = 27
4320:where
4267:β3) =
4259:where
3848:). If
3795:where
3640:) ) =
3416:, and
2010:by log
1585:
1582:
1579:
722:googol
489:googol
446:Cosmos
235:, and
84:, and
52:, and
12155:Names
11957:BH(3)
11835:order
11583:arXiv
11546:(PDF)
11531:(PDF)
11419:S2CID
10918:) = (
10863:-1 β
10830:+1 =
10676:= 4,
10573:β₯ 2,
10501:) = 2
10441:+ 1:
10130:table
10116:65533
10104:65536
9147:; if
8011:200.3
7878:200.3
4824:2 = 4
4742:with
4606:=(10β
4529:When
4421:. If
1226:4.829
1191:4.829
469:Sagan
211:atoms
164:6.022
119:cells
105:, 1.0
11677:<
11591:ISBN
11500:ISSN
11470:PMID
11411:PMID
11403:ISSN
11155:and
11117:OEIS
11093:The
11044:) =
10930:β 1)
10892:) =
10839:(1))
10782:) =
10590:) =
10569:for
10545:for
10526:) =
10490:) =
10466:) =
10455:) =
10437:) =
9720:and
9601:and
9527:and
9460:and
9426:and
9303:and
8570:<
8470:<
8039:<
7649:2).
7131:) =
6185:9.73
6051:3.81
5775:1.10
5324:1.10
5262:1.26
5118:1.49
4352:) =
4324:and
4074:<
4052:<
3980:<
3956:<
2885:= (
2776:and
2716:9.73
2356:and
2302:9.73
2239:9.73
2169:and
2112:3.81
2047:7.21
1838:and
1739:and
1382:<
1376:<
1332:<
1307:<
1264:and
900:any
898:cite
623:6000
596:3003
254:and
194:3.6)
190:(5.3
156:mole
150:The
130:bits
11460:PMC
11450:doi
11395:hdl
11385:doi
11171:of
11078:+1
11027:+1
10937:Ο2+
10776:Ο+1
10730:+ 2
10662:(=
10649:Ο+1
10549:β₯ 2
10474:= 2
9951:log
9745:log
9729:log
9667:log
9651:log
9206:6.2
9040:2.3
9035:100
8965:100
8867:2.3
8862:100
8811:100
8719:2.3
8714:100
8669:100
8615:100
8467:2.3
8462:100
8412:200
8342:100
8321:2.3
8316:100
8285:200
8234:100
8213:2.3
8208:100
8183:200
8138:100
8084:100
8036:2.3
7975:200
7924:100
7903:2.3
7848:200
7806:0.3
7798:200
7753:100
7728:200
7700:100
7677:200
7660:100
7584:100
7579:100
7574:100
7517:100
7512:100
7470:100
7349:...
7311:...
6686:154
6177:497
5875:4.3
5733:3.6
5577:4.3
5550:4.3
5518:4.3
5484:536
5365:100
5297:024
5291:640
5285:334
5279:638
5141:047
5135:862
5110:933
5104:589
5063:4.3
5036:728
5019:2.0
5011:536
4112:in
4108:If
2708:497
2630:7.3
2523:2.8
2294:497
2231:497
1759:534
1727:533
1695:4.3
1690:533
1659:728
1634:531
1594:536
1451:100
1422:100
1401:.)
1365:if
1109:).
911:by
870:MIT
825:964
740:100
684:933
678:589
610:or
563:600
550:or
536:303
504:100
473:Cf.
471:.
319:1.1
175:DNA
12374::
11831:in
11772:.
11748:.
11724:.
11674:10
11661:10
11639:.
11589:.
11539:41
11533:.
11494:.
11468:.
11458:.
11446:13
11444:.
11440:.
11417:.
11409:.
11401:.
11393:.
11381:40
11379:.
11375:.
11248:.
11137:,
11082:s)
11080:n'
11070:β
11062:β
11031:s)
11029:n'
11019:β
11011:β
10987:β
10983:β
10979:β
10969:Ο3
10958:β
10954:β
10950:β
10926:β
10922:β
10914:β
10910:β
10897:Ο+
10886:Ο2
10875:β
10871:β
10859:β
10846:Ο+
10826:β
10818:β
10797:β
10761:β
10753:β
10728:+1
10684:+1
10667:64
10619:,
10559:+1
10470:+
10110:10
10072:10
10028:10
9955:10
9940:10
9935:10
9920:10
9912:10
9897:10
9880:10
9805:.
9766:10
9761:10
9749:10
9733:10
9708:10
9692:10
9688:10
9683:10
9671:10
9655:10
9615:10
9610:10
9585:10
9581:10
9576:10
9541:10
9536:10
9507:10
9502:10
9469:10
9442:10
9396:90
9388:10
9367:10
9363:10
9353:10
9312:10
9289:10
9285:10
9246:50
9242:10
9213:10
9200:10
9169:10
9130:10
9119:10
9100:10
9075:10
9060:10
9024:10
9016:98
9005:10
8980:10
8938:10
8927:10
8908:10
8899:10
8887:10
8851:10
8843:98
8832:10
8823:10
8784:10
8773:10
8754:10
8739:10
8703:10
8695:98
8684:10
8641:10
8573:10
8562:10
8551:10
8526:10
8511:10
8473:10
8451:10
8426:10
8408:10
8393:98
8382:10
8357:10
8305:10
8296:10
8281:10
8266:98
8255:10
8246:10
8197:10
8179:10
8164:98
8153:10
8110:10
8042:10
8020:10
7989:10
7971:10
7939:10
7887:10
7862:10
7844:10
7818:10
7794:10
7768:10
7724:10
7713:10
7673:10
7647:10
7621:24
7617:10
7606:10
7601:10
7588:12
7547:24
7543:10
7532:10
7521:12
7487:24
7483:10
7474:12
7376:64
7363:62
7356:63
7337:10
7251:10
7246:10
7239:10
7235:10
7226:10
7218:10
7196:10
7192:10
7159:10
7152:10
7148:10
7139:10
7117:10
7113:10
7072:10
7061:10
7052:10
7046:10
7034:10
6993:10
6978:10
6937:10
6922:10
6881:10
6866:10
6828:10
6813:10
6775:10
6760:10
6722:10
6707:10
6670:10
6651:10
6581:10
6566:10
6525:10
6514:10
6505:10
6499:10
6487:10
6449:10
6434:10
6396:10
6381:10
6343:10
6328:10
6290:10
6275:10
6237:10
6222:10
6196:32
6192:10
6166:10
6147:10
6106:10
6091:10
6062:17
6058:10
6047:10
6042:10
6035:10
6018:10
5994:11
5978:10
5937:10
5922:10
5859:10
5759:10
5744:12
5740:10
5729:10
5640:10
5625:10
5613:10
5609:10
5604:10
5599:10
5561:10
5534:10
5527:10
5502:10
5478:65
5378:10
5361:10
5356:10
5308:10
5269:10
5172:10
5155:10
5150:10
5129:24
5125:10
5098:82
5047:10
5030:19
5026:10
5005:65
4765:10
4670:10
4631:10
4618:10
4588:10
4556:10
4483:10
4105:.
4082:65
4060:64
4018:65
3998:65
3992:10
3986:10
3971:65
3931:10
3926:10
3919:10
3915:10
3906:10
3898:10
3877:10
3871:10
3827:10
3739:10
3726:10
3687:10
3682:10
3671:10
3656:10
3648:10
3622:10
3544:10
3538:10
3529:10
3516:10
3478:10
3463:10
3457:10
3448:10
3433:10
3424:10
3388:10
3363:10
3354:10
3149:10
3095:10
3073:10
2981:10
2893:β
2889:β
2849::
2838:.
2790:10
2784:10
2758:10
2752:10
2727:32
2723:10
2697:10
2678:10
2669:10
2658:.
2637:10
2621:10
2595:10
2565:10
2551:.
2534:12
2530:10
2504:10
2475:10
2426:10
2397:10
2364:10
2338:10
2313:32
2309:10
2283:10
2264:10
2250:32
2246:10
2220:10
2211:10
2205:10
2177:10
2151:10
2123:17
2119:10
2108:10
2103:10
2096:10
2089:10
2054:10
2038:10
2023:10
2012:10
1978:10
1942:10
1925:10
1898:10
1891:10
1846:10
1820:10
1798:10
1753:65
1747:10
1721:65
1715:10
1684:65
1673:10
1653:19
1649:10
1628:65
1617:10
1588:65
1460:10
1435:10
1418:10
1413:10
1385:10
1335:10
1313:10
1298:10
1272:10
1246:10
1210:10
1187:10
1182:10
1177:10
1172:10
1167:10
1130:10
1087:10
1034:10
995:10
974:10
962:10
821:10
816:10
811:10
784:34
780:10
775:10
770:10
736:10
731:10
718:10
709:=
672:82
659:=
652:10
619:10
592:10
583:=
559:10
532:10
523:=
500:10
491:=
479:.
475:,
435:.
406:.
315:10
310:10
305:10
300:10
295:10
231:,
198:10
168:10
138:TB
80:,
76:,
48:,
44:,
12194:e
12187:t
12180:v
11814:e
11807:t
11800:v
11783:.
11758:.
11734:.
11699:3
11694:1
11691:+
11688:n
11680:3
11669:n
11597:.
11585::
11555:.
11512:.
11476:.
11452::
11425:.
11397::
11387::
11230:1
11222:=
11217:c
11189:c
11107:n
11103:n
11076:n
11072:n
11068:n
11064:n
11060:n
11056:n
11054:(
11051:n
11049:Ο
11046:f
11042:n
11040:(
11038:Ο
11035:f
11025:k
11021:n
11017:n
11013:n
11009:n
11005:n
11003:(
11000:k
10998:Ο
10995:f
10991:)
10989:n
10985:n
10981:n
10977:n
10973:n
10971:(
10966:f
10962:)
10960:k
10956:n
10952:n
10948:n
10944:n
10942:(
10939:k
10934:f
10928:n
10924:n
10920:n
10916:n
10912:n
10908:n
10904:n
10902:(
10899:n
10894:f
10890:n
10888:(
10883:f
10879:)
10877:k
10873:n
10869:n
10865:k
10861:n
10857:n
10853:n
10851:(
10848:k
10843:f
10836:k
10832:g
10828:k
10824:n
10820:k
10816:n
10812:n
10810:(
10807:k
10803:g
10799:n
10795:n
10791:n
10789:(
10787:Ο
10784:f
10780:n
10778:(
10773:f
10769:)
10763:n
10759:n
10755:n
10751:n
10747:n
10743:n
10741:(
10739:Ο
10736:f
10726:k
10722:g
10717:k
10713:g
10711:(
10709:Ο
10706:f
10702:n
10698:n
10696:(
10694:Ο
10691:f
10682:k
10678:g
10674:0
10671:g
10664:g
10656:Ο
10653:f
10646:f
10640:Ο
10637:f
10629:A
10625:n
10621:n
10617:n
10615:(
10613:A
10609:n
10605:n
10601:n
10599:(
10596:n
10592:f
10588:n
10586:(
10584:Ο
10581:f
10575:k
10571:n
10567:n
10563:n
10561:(
10557:k
10553:f
10547:n
10543:n
10539:n
10535:n
10533:(
10531:2
10528:f
10524:n
10522:(
10520:3
10517:f
10513:)
10507:n
10503:n
10499:n
10497:(
10495:1
10492:f
10488:n
10486:(
10484:2
10481:f
10476:n
10472:n
10468:n
10464:n
10462:(
10460:0
10457:f
10453:n
10451:(
10449:1
10446:f
10439:n
10435:n
10433:(
10431:0
10428:f
10417:k
10401:k
10397:f
10376:)
10373:n
10370:(
10365:n
10361:f
10357:=
10354:)
10351:n
10348:(
10339:f
10318:)
10315:n
10312:(
10307:n
10302:1
10296:k
10292:f
10288:=
10285:)
10282:n
10279:(
10274:k
10270:f
10256:n
10252:n
10238:)
10235:n
10232:(
10227:n
10222:0
10218:f
10214:=
10211:)
10208:n
10205:(
10200:1
10196:f
10185:n
10183:(
10171:)
10168:n
10165:(
10160:0
10156:f
10132:.
10098:2
10076:n
10063:n
10059:2
10032:n
10019:n
10015:n
9994:n
9963:a
9947:+
9944:b
9931:=
9924:b
9916:a
9908:=
9901:b
9890:)
9884:a
9876:(
9852:X
9848:X
9844:X
9840:X
9836:X
9832:X
9828:X
9786:9
9783:=
9780:)
9777:)
9770:9
9757:(
9741:(
9705:=
9702:)
9699:)
9679:(
9663:(
9619:9
9545:b
9511:a
9473:b
9446:a
9428:b
9424:a
9393:=
9385:1
9377:1
9374:=
9357:9
9344:1
9316:9
9217:3
9189:n
9173:n
9149:n
9135:1
9126:)
9116:(
9111:8
9107:)
9097:(
9092:2
9086:n
9082:)
9072:(
9069:=
9066:n
9031:)
9021:(
9012:)
9002:(
8997:2
8991:n
8987:)
8977:(
8974:=
8971:n
8955:)
8943:1
8934:)
8924:(
8919:8
8915:)
8905:(
8896:=
8893:3
8858:)
8848:(
8839:)
8829:(
8820:=
8817:3
8801:)
8789:1
8780:)
8770:(
8765:8
8761:)
8751:(
8748:=
8745:2
8710:)
8700:(
8691:)
8681:(
8678:=
8675:2
8659:)
8647:n
8621:n
8605:n
8591:)
8588:1
8585:+
8582:n
8579:(
8567:1
8558:)
8548:(
8543:2
8537:n
8533:)
8523:(
8520:=
8517:n
8491:)
8488:1
8485:+
8482:n
8479:(
8458:)
8448:(
8443:2
8437:n
8433:)
8423:(
8420:=
8417:)
8401:2
8398:(
8389:)
8379:(
8374:2
8368:n
8364:)
8354:(
8351:=
8348:n
8312:)
8302:(
8293:=
8290:)
8274:2
8271:(
8262:)
8252:(
8243:=
8240:3
8204:)
8194:(
8191:=
8188:)
8172:2
8169:(
8160:)
8150:(
8147:=
8144:2
8128:)
8116:n
8090:n
8074:n
8060:)
8057:1
8054:+
8051:n
8048:(
8031:n
8027:)
8017:(
8014:=
8006:1
8000:n
7996:)
7986:(
7983:=
7980:)
7964:2
7961:(
7956:2
7950:n
7946:)
7936:(
7933:=
7930:n
7898:4
7894:)
7884:(
7881:=
7873:3
7869:)
7859:(
7856:=
7853:)
7837:2
7834:(
7829:2
7825:)
7815:(
7812:=
7809:)
7803:+
7787:2
7784:(
7779:2
7775:)
7765:(
7762:=
7759:4
7717:2
7709:=
7706:3
7669:=
7666:2
7626:+
7610:2
7536:2
7528:=
7479:=
7418:/
7414:/
7373:g
7360:g
7353:g
7344:9
7341:g
7334:g
7325:8
7322:g
7318:9
7315:g
7306:3
7303:g
7299:3
7296:g
7292:4
7289:g
7283:2
7280:g
7276:2
7273:g
7269:3
7266:g
7178:2
7175:g
7077:1
7068:)
7058:(
7055:=
7043:=
7040:2
7012:=
7009:1
7004:9
7000:)
6990:(
6987:=
6984:9
6956:=
6953:1
6948:8
6944:)
6934:(
6931:=
6928:8
6900:=
6897:1
6892:7
6888:)
6878:(
6875:=
6872:7
6844:1
6839:6
6835:)
6825:(
6822:=
6819:6
6791:1
6786:5
6782:)
6772:(
6769:=
6766:5
6738:1
6733:4
6729:)
6719:(
6716:=
6713:4
6681:3
6677:)
6667:(
6662:2
6658:)
6648:(
6625:4
6619:4
6597:1
6592:3
6588:)
6578:(
6575:=
6572:3
6556:)
6550:1
6547:g
6530:1
6521:)
6511:(
6508:=
6496:=
6493:2
6465:1
6460:9
6456:)
6446:(
6443:=
6440:9
6412:1
6407:8
6403:)
6393:(
6390:=
6387:8
6359:1
6354:7
6350:)
6340:(
6337:=
6334:7
6306:1
6301:6
6297:)
6287:(
6284:=
6281:6
6253:1
6248:5
6244:)
6234:(
6231:=
6228:5
6201:)
6182:(
6173:)
6163:(
6158:2
6154:)
6144:(
6122:1
6117:4
6113:)
6103:(
6100:=
6097:4
6029:2
6025:)
6015:(
5989:2
5985:)
5975:(
5953:1
5948:3
5944:)
5934:(
5931:=
5928:3
5870:4
5866:)
5856:(
5850:7
5844:2
5841:=
5826:2
5822:2
5817:2
5812:2
5807:2
5802:2
5797:2
5770:4
5766:)
5756:(
5724:3
5717:5
5711:3
5708:=
5697:3
5693:3
5688:3
5683:3
5678:3
5656:1
5651:4
5647:)
5637:(
5634:=
5631:4
5622:=
5572:3
5568:)
5558:(
5555:=
5545:2
5541:)
5531:(
5513:2
5509:)
5499:(
5495:2
5481:,
5474:2
5469:2
5465:=
5462:6
5456:2
5453:=
5440:2
5436:2
5431:2
5426:2
5421:2
5416:2
5394:2
5389:3
5385:)
5375:(
5372:=
5341:k
5319:3
5315:)
5305:(
5294:,
5288:,
5282:,
5276:,
5273:3
5256:4
5250:3
5247:=
5238:3
5234:3
5229:3
5224:3
5207:.
5183:2
5179:)
5169:(
5166:=
5138:,
5132:,
5107:,
5101:,
5094:M
5058:2
5054:)
5044:(
5033:,
5008:,
5001:2
4997:=
4994:5
4988:2
4985:=
4974:2
4970:2
4965:2
4960:2
4955:2
4864:2
4860:2
4855:2
4850:2
4806:n
4788:n
4784:p
4779:)
4773:n
4762:(
4752:k
4748:q
4744:q
4722:k
4719:q
4715:m
4707:k
4704:q
4700:f
4687:n
4673:)
4667:(
4661:n
4657:f
4645:n
4626:n
4608:n
4592:n
4577:n
4573:n
4559:)
4553:(
4547:n
4543:f
4531:k
4524:n
4506:n
4502:p
4497:)
4491:n
4480:(
4470:k
4452:k
4448:m
4440:k
4436:f
4423:n
4409:)
4406:1
4403:(
4398:n
4393:1
4387:k
4383:f
4379:=
4376:)
4373:n
4370:(
4365:k
4361:f
4350:k
4348:β
4346:n
4342:g
4338:k
4334:f
4330:k
4326:m
4322:k
4308:)
4305:n
4302:(
4297:m
4292:k
4288:f
4277:h
4273:n
4269:g
4265:m
4261:m
4247:)
4244:n
4241:(
4236:m
4232:g
4221:g
4207:)
4204:1
4201:(
4196:n
4192:f
4188:=
4185:)
4182:n
4179:(
4176:g
4166:f
4162:f
4158:m
4154:n
4150:n
4136:)
4133:n
4130:(
4125:m
4121:f
4110:m
4093:)
4090:1
4087:(
4078:f
4071:)
4068:4
4065:(
4056:f
4049:G
4029:)
4026:1
4023:(
4014:f
4010:=
4007:)
4004:2
3983:(
3977:2
3965:3
3959:3
3953:G
3895:=
3892:)
3889:2
3883:3
3868:(
3858:m
3854:f
3850:n
3836:)
3831:5
3820:3
3817:(
3812:2
3808:f
3797:m
3783:)
3780:n
3777:(
3772:m
3768:f
3757:f
3753:n
3734:n
3723:=
3720:)
3717:n
3714:(
3711:f
3675:5
3664:3
3626:5
3615:3
3589:n
3574:n
3567:n
3553:)
3550:n
3535:(
3532:=
3524:n
3502:x
3498:x
3484:3
3472:)
3469:1
3466:+
3454:(
3445:=
3442:)
3439:3
3430:(
3404:a
3399:6
3395:)
3385:(
3382:=
3379:a
3374:5
3370:)
3360:(
3344:n
3323:1
3320:+
3317:n
3313:k
3308:)
3303:1
3300:+
3297:n
3293:(
3270:1
3267:+
3264:n
3260:k
3235:n
3231:k
3206:n
3202:k
3190:n
3172:n
3168:k
3163:)
3157:n
3146:(
3136:b
3118:1
3113:3
3109:)
3103:2
3092:(
3089:=
3086:3
3081:3
3050:1
3047:=
3040:b
3018:b
3004:b
2999:3
2995:)
2989:2
2978:(
2955:b
2950:k
2946:)
2940:n
2932:a
2929:(
2919:n
2915:b
2909:)
2907:b
2903:n
2899:a
2895:n
2891:b
2887:a
2873:b
2868:n
2860:a
2824:n
2796:5
2764:4
2732:)
2713:(
2704:)
2694:(
2689:2
2685:)
2675:(
2646:)
2641:6
2627:(
2601:)
2592:(
2571:)
2562:(
2539:)
2520:(
2515:3
2511:)
2501:(
2481:)
2472:(
2452:1
2432:)
2423:(
2403:)
2394:(
2382:)
2370:5
2344:4
2318:)
2299:(
2290:)
2280:(
2275:2
2271:)
2261:(
2258:=
2255:)
2236:(
2227:)
2217:(
2195:)
2183:3
2157:2
2063:)
2058:8
2044:(
2019:x
2015:x
2008:x
1994:x
1989:n
1985:)
1975:(
1965:x
1946:x
1936:n
1932:)
1922:(
1919:=
1914:x
1909:n
1905:)
1895:(
1864:)
1861:1
1858:+
1855:n
1852:(
1826:n
1794:g
1790:o
1787:l
1777:n
1771:)
1756:,
1724:,
1687:,
1680:)
1670:(
1664:)
1656:,
1642:6
1639:(
1631:,
1624:)
1614:(
1604:2
1591:,
1553:2
1547:.
1541:.
1535:.
1527:2
1520:2
1510:=
1507:4
1501:2
1476:2
1471:3
1467:)
1457:(
1454:=
1446:2
1442:)
1432:(
1429:=
1379:a
1373:1
1353:)
1350:1
1347:+
1344:n
1341:(
1329:a
1324:n
1320:)
1310:(
1304:n
1278:6
1252:5
1221:5
1217:)
1207:(
1204:=
1146:a
1141:n
1137:)
1127:(
1091:n
1083:=
1080:)
1077:n
1074:(
1071:f
1045:n
1041:)
1031:(
1011:1
1006:n
1002:)
992:(
989:=
986:2
980:n
971:=
968:n
938:)
932:(
927:)
923:(
919:.
905:.
862:.
727:=
692:1
681:,
675:,
668:2
650:Γ
646:Γ
239:.
196:Γ
192:Β±
166:Γ
123:Γ
107:Γ
95:Γ
34:.
20:)
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