Notable Properties of Specific Numbers
The number centillion, which for a long time was the largest number with a single-word name in English (and many other languages that use the Chuquet names). In the majority of the English-speaking world centillion meant 10600. The Oxford English Dictionary  gives a usage from 1852, almost 100 years before the invention of the name googolplex. See also vigintillion, millillion and milli-millillion.
This is (approximately) the maximum value that can be represented in the commonly-used IEEE 754 double-precision (1+11+52 bit) floating-point format.
(the actual value of the lower "Skewes' number")
This is the first point at which the prime counting function pi(x) exceeds the logarithmic integral li(x). This is the quantity that was originally estimated by an upper bound of eee79 ≈ 10101034, the "Skewes number".
The approximation 1.39822×10316 was given by Bays & Hudson in 2000, then Chao & Plymen improved it somewhat in 2005, and finally Demichel (later in 2005) improved the estimate to 1.397162914×10316.
See also 906150257.
(the te Riele revision of "Skewes' number")
(the "single-perturbation count")
If you chose a random moment in the universe's history, then chose a random particle in the universe and moved the particle to a random location somewhere else in the universe at that moment, you would have a total of 1.41×10408 distinct choices (as limited by the uncertainty principle). The formula for this is equivalent to the universe's 4-dimensional volume times the age times the number of particles. (Moving a particle instantly by a large distance would usually violate special relativity by exceeding the speed of light, but "particles" moving at faster than the speed of light must be considered when modeling physics by one of the gauge theories such as Quantum Electrodynamics.)
This is the single-perturbation count. Its factorial gives the total number of complete shufflings of the entire known universe's history. The exact value is really arbitrary, not only because of the choices of age and size (due to cosmic inflation, there is much that is beyond our event horizon and can never be seen), but also because I didn't adjust for the curvature of space-time on large scales, which throws off any such calculation quite a bit.
In the Lalitavistara, a biography of Gautama Buddha which was written mainly during the first couple centuries A.D., Gautama is asked to name the powers of ten starting with koti, which is 107. He gives names for powers of ten up to the tallaksana, 1053. He then describes successive "numerations", the dvajagravati=1099, the dvajagranisamani=10145, and several more culminating in 10421, which is given the name uttaraparamanurajahpravesa28. There are many other stories like this in the culture of India from that period, which shows the extent to which they were interested in large numbers, and also helps explain their need to use place-value notation with a symbol for zero.29
According to , this is a "popular estimate" of the number of "vacua" in the string theory landscape within eternal chaotic inflation models. In such cosmologial models, our universe is one of many that were produced (and indeed are still being produced) through a creation process predicted by physics string theory.
1.7989021000...×10571 = 20!×319 × 30!×229 × (60!/(5!12))6 × (60!)2 / 210
This is the number of ways to arrange the pieces on a Teraminx, a dodecahedron-shaped combinatorial puzzle with 530 movable pieces. Teraminx is to the Megaminx as the 7x7x7 cube is to the normal 3x3x3 Rubik's Cube. These puzzles have been made by dedicated hobbyists with access to CAD-CAM prototyping machines. Here is a video of one being assembled, and here you can see the puzzle being manipulated.
The value of centillion in the long scale system; see 10303.
Randall Munroe made this estimate of the number of possible 140-character Twitter messages, where you're allowed to use any Unicode character, and accounting for the complicated way Twitter counts characters.
A 6-state turing machine, found by Heiner Marxen and Juergen Buntrock in 2001 March, takes 3.00233×101730 steps before halting with 1.29149×10865 ones on the tape. It was a busy beaver record holder for a while. Using a very small set of state transition rules, it iterates X'=2K×X several times in a row, with a chaotic deterministic low-probability exit condition. Its record was broken by Terry and Shawn Ligocki, whose machine leaves 4.6×101439 marks.
4.57936...×10917 = 215×310×56×75×114×...×2039×2053×2063
The smallest number that has at least a googol distinct factors. Its exact value is 457936006084633875 260691932542213506579481395376 080192442872707759996212114957 373537195900697943283211344130 969977204683723647091975242566 556807073476262370119366712949 612051508874565615465951982148 103948322515169952026557331614 199239782652240565877185274882 891122589783986489974588207230 026310073238799349251084594897 863556829085566422093207975001 895285824382289647389848615424 710629561529529589935914349946 023950287863307022313442880758 800532983282085207377266536998 146723331964258315488766981883 904240306133944424567760471103 539279962416731476757145320641 439420037963516042879919957607 890943287019373144639492683640 803862704805497501551907216898 677744138585826270309663329962 841518933729157858558919253022 063551926057138672786596389094 200184031909805595086778342937 081605771699885426749776777391 919555685119629369584896777148 250878775274042686107865894781 763500774758450843791837394393 056896301600021929961984000000. The factorization of this number, along with the other record setters up to 103535, was found by Achim Flammenkamp7. See also 12, 840, 45360, 720720, 3603600, 245044800, 278914005382139703576000 and 2054221614063184107682218077003539824552559296000.
7.7263039555...×10992 = 20!×319 × 30!×229 × (60!/(5!12))12 × (60!)3 / 217
This is the number of ways to arrange the pieces on a Petaminx, a dodecahedron-shaped combinatorial puzzle with 950 movable pieces. Petaminx is to the Megaminx as a 9x9x9 cube would be to the normal 3x3x3 Rubik's Cube, if such a thing existed. Believe it or not, someone actually built this puzzle using parts cast from an industrial prototyping machine, and sold it online for over $3000 U.S. A video of the puzzle being used can be seen here.
1.97231222789×101015 = 2172395117511111317217111912313291331737172411143147135313
This is the Gödel number of the smallest theorem in the formal system P used by Gödel in his first Incompleteness theorem. The smallest theorem in P is "0=0". This has only 3 symbols, but the symbol '=' is not a basic sign and must be expanded first before deriving the Gödel number. The expanded form of "0=0" is a2 ∀ (~(a2(0)) ∨ a2(0)). This formula has 16 basic signs, with individual Gödel numbers 172, 9, 11, 5, 11, 172, 11, 1, 13, 13, 7, 172, 11, 1, 13, 13. To get the Gödel number of the formula these numbers are used as the exponents of the first n prime numbers, where n is the number of basic signs.
6.690926087×101054 = 28 × (8!/2)×37 × (12!/2)×211 × (24!/2)63 / ((246)/2)56
The number of ways to arrange a 17×17×17 Rubik's Cube. As of late 2014, this remains the largest N×N×N cube ever made as a real physical object. It has 1539 parts and at least two copies have been made via the 3D printing service shapeways and would cost well over $10,000 US. Videos of it can be seen here (inventor Oskar van Deventer) and here (cubist/reviewer RedKB).
(Lehman's 1966 estimate of Skewes' number)
In 1966, Lehman gave the first estimate of the actual value of the first point at which the prime counting function π(x) exceeds the logarithmic integral li(x), stating that it was somewhere between 1.53×101165 and 1.65×101165. This was the first published computed estimate, and improved significantly on Skewes' estimate 10101034. See also 1.397×10316.
M4253, the 19th Mersenne Prime, and the subject of an interesting debate about the nature of discovery. In 1961 Alexander Hurwitz designed and ran a program to search for Mersenne primes on an IBM 7090 computer. The computer program found this number and quite a while later found M4423. Because of the way the computer's output was stacked, Hurwitz saw (and therefore "discovered") the larger of the two primes first. This raises the question first posed by Hurwitz's colleague John Selfridge: Can the primes be considered to have been discovered when the program finished calculating them, or does "discovery" not happen until a human observes it? Hurwitz replied, "Forgetting about whether the computer 'knew', what if the computer operator who piled up the output looked?"
In 2002, M.-Ch. Liu and T. Wang improved on Chen and Wang's 1989 result (1043000) showing that the weak Goldbach conjecture is true for all numbers larger than this. See 3.248...×106846168. (Some sources give "2×101346" rather than "e3100".)
A former record-holder for the 6-state busy beaver Turing machine takes about 2.5×102879 steps before halting with 4.6×101439 ones on the tape. The machine was discovered by Terry and Shawn Ligocki in 2007, and overtook a Marxen-Buntrock machine that left 1.3×10865 marks. The machine that surpassed it leaves 3.515×1018267 marks. See also 107 and 47176870.
Number of steps taken by a certain 6-state, 5-tuple Turing machine before halting. It was a record-holder for 5 years, and was found by Buntrock and Marxen in 2000. The record was broken by Terry and Shawn Ligoki in December 2007, see 2.5×102879. See 107 for more.
An "11-fold multiperfect number" found by Woltman on 2001 Mar 13th. The sum of its divisors (including 1 and itself) is exactly 11 times its value. See also 120, 496, 30240, and 154345556085770649600.
Lower bound for the number of steps a 6-state, 5-tuple Turing machine can take, on an initially blank tape, before halting, found by Terry and Shawn Ligoki in December 2007. It supplants the previous record belonging to a Marxen-Buntrock machine, which took 3×101730 steps. See 107 and 47176870 for more.
This number is often called millillion, following the pattern established by Chuquet (see my table of standard names) and extended by others to such names as decillion, vigintillion and centillion. The name probably originated with Henkle as published by Brooks in 1904; see this discussion; however the name is an obvious parallel to vigintillion and centillion to anyone who knows how to count in Latin.
See also 103000003.
The value of the number called zài in one ancient Chinese system for naming large numbers36. In this system, The successive names yì, zhào, etc. name successive squares of wàn (which is 104), thus yì=108, zhào=1016, and so on up to zài=104096. In modern usage, zài is "merely" 1044. In the Knuth -yllion naming system, 104096 is one decyllion; in the more mainstream Conway-Wechsler system, it is ten milliquattuorsexagintatrecentillion.
This is a 4181-digit number, and is prime, and looks like:
with the same number of 000000's on both sides. This makes it a palindrome, and since its digits are all 0, 1, or 8, it is strobogrammatic (looks the same upside down). (from Jason Earls via Prime pages)
See also 1.000...×10101.
104594+3×102297+1 is a large palindromic prime discovered by Harvey Dubner in 1987. As you can tell from the formula, its digits consist of a 1 followed by (4594-2297-1)=2296 zeros, followed by a 3, followed by another 2296 zeros, and then a 1. Because it is prime, it could be used to construct a very large Smith number.
This is (approximately) the maximum value that can be represented in several implementations of IEEE 754 extended double floating-point formats, and the IEEE 754r "binary128" format. They all have a 15-bit exponent field. In most other respects, the various extended double formats differ. The most common is exemplified by the Intel IA-64 architecture's 10-byte "extended double-precision" which has a 63-bit mantissa; less common is the 16-byte "quadruple precision" (such as found on Digital VAX and Alpha systems) with a 112-bit mantissa. The IEEE 754 specifications for "extended" formats allow the implementer to choose pretty nearly any exponent and mantissa size they want.
(current limit for deterministic prime_p tests)
As of 2007, this is the approximate limit on the size of numbers that can be shown to be prime or composite using deterministic primality tests such as the elliptic curve method. Such tests determine for certain whether a number is prime or composite. It takes a 3 GHz processor about a month to prove primeness of a 5000-digit number, using the ECPP (Elliptic Curve Primality Proving) method46. See also 1015000 and 1.7505×1020561.
This is one of many Mersenne primes discovered by computer using the Lucas-Lehmer test. The Lucas-Lehmer test states that you can test a Mersenne number Mn for primeness by computing the sequence S1 = 4, Sn+1 = Sn2 - 2, and checking if Sn-1 divides evenly into Mn. If it does, Mn is prime (see A003010 for examples). The sequence starts: 4, 14, 194, 37634, 1416317954, ... and grows very quickly, doubling in digits each time.
This test can be programmed on a computer using binary arithmetic and requires no division (the modulo test can be performed by a process similar to casting out nines but in base 2n). The result is that today, nearly anyone with a home computer now has a shot at discovering Mersenne primes. In 1978, two high-school students (Noll & Nickel) discovered 221701-1 on a local university mainframe computer, and by the late 1990's all new Mersenne primes were being discovered by individual personal computers.
106572+3×103286+1 is a large palindromic prime discovered by Harvey Dubner in 1990. As you can tell from the formula, its digits consist of a 1 followed by (6572-3286-1)=3285 zeros, followed by a 3, followed by another 3285 zeros, and then a 1. Because it is prime, it could be used to construct a very large Smith number.
This is the product 1!×2!×3!×...×98!×99!×100!. It is not a perfect square, but if you remove one of the factorials from the product, the result is a perfect square. Which one should be removed? Try to figure it out yourself or see 210691031040000 for the answer and an explanation.
A very rough estimate of the number of possible chess games, assuming both sides are conspiring to draw out the game as long as possible (see 8848). This is approximately 268848, based on the common estimate of 30 choices per two plies, less a few because in this scenario the players are constrained to moves that avoid a pawn move or capture. For competitive games (in which at least one player is actually trying to win), something like Shannon's estimate of 10120 is much more realistic.
See also 26830.
(current limit for probabilistic prime_p tests)
As of 2007, this is the approximate limit on the size of numbers that can be shown to be composite using probabilistic primality tests. Such tests show that a number is either composite, or very probably prime (i.e. with probability a tiny bit less than 1.000).
See also 105000.
5.19344195...×1015070 = 26384405 + 44052638
This number was found by Greg Childers, and shown prime using a deterministic method by Franke, Kleinjung, Morain & Wirth. It is well beyond the normal limit for deterministic prime testing, and as Leyland states, such numbers are good for testing deterministic prime test methods because they do not allow for convenient "shortcuts" (like the twin primes and Mersenne primes do).
2.2557375222255737522...×1015599 = (2255737522 × R15600) / 1111111111 + 1
In 2002 Harvey Dubner and David Broadhurst40 showed that this number is prime. It is of interest because all of its digits are also prime (being either 2, 3, 5 or 7). It is the largest known number with this property. R15600 is the repunit with 15600 digits; note that R15600/R10 = 100000000010000000001000...00001, a 15591-digit number. As a result, when multiplied by 2255737522 the result simply consists of the digits 2255737522 repeated 1560 times (then we add 1 to make the last digit a 3). Dubner also demonstrated the primeness of the slightly smaller (2255725272 × R15600) / 1111111111 + 1.
As of June 2010, the record for the 6-state busy beaver Turing machine takes about 7.412×1036534 steps before halting with 3.515×1018267 ones on the tape. The machine was discovered by Pavel Kropitz in June 2010, and overtook a Ligocki machine that left 4.6×101439 marks. See also 107 and 47176870.
The Ackermann function
The Ackermann function is a function that grows very fast, but has a surprisingly innocent-looking definition. Using the two-argument version of Peters, A(m,n) = n+1 (if m=0) or A(m-1,1) (if m>0 and n=0):> or A(m-1,A(m,n-1)) <$:(for remaining cases). This function produces the following table:
The first row is the positive integers, and each subsequent row is an Nth-term sequence generated from the row before it. Row 2 is linear, row 3 is exponential, and row 4 grows like the higher hyper4 operator.
1.7505...×1020561 = (((((((((23+3)3+30)3+6)3+80)3+12)3+450)3+894)3+3636)3+70756)3+97220
As of 2009, this was the largest number to be proven prime via the general-purpose ECPP algorithm. The work was distributed amongst a large number of computers, taking nearly a year and an aggregate computing time equivalent to a single 2.4-GHz Opteron running for over 6 years46. The sequence: 2, 23+3, (23+3)3+30, ... is Sloane's A051254 and is related to the problem of finding and proving the value of Mills' constant.
See also 105000.
The "largest known easy-to-remember prime", discovered by the "Amdahl six", a team of large prime hunters. They discovered this 21078-digit prime number as part of a larger project to identify large primes fitting the pattern p = A 2B +/- 1. It can be remembered by its formula 235235 × 270000 - 1. Notice the repetition of the 2, 3 and 5: the first 3 prime numbers; the next prime 7 is the first digit of the exponent. See also 1.23456...×1019.
For a while this was the largest known prime number of the form xy + yx (where x and y are integers greater than 1), i.e. a prime that is also a Leyland number. It was later superceded by 8.37300×1030007.
1028572+8×1014286+1 is a large palindromic prime discovered by Daniel Heuer in 2001. As you can tell from the formula, its digits consist of a 1 followed by (28572-14286-1)=14285 zeros, followed by an 8, followed by another 14285 zeros, and then a 1. Because it is prime, it could be used by Costello  to construct a record Smith number.
This is (approximately) the maximum value that can be represented in the double-precision format on the Burroughs 6x00 family of mainframe computers, and is the highest overflow value for any hardware floating-point format I have heard of to date. Numerous software-implemented formats exceed it.
As of late 2014, this is the largest known prime number of the form xy + yx (where x and y are integers greater than 1). Large primes of this type have been extensively studied by Paul Leyland, and such numbers are called "Leyland primes" in his honor. The more general case a number of the form xy + yx (without the need to be prime) is called a Leyland number. Numberphile has a video about it here: Leyland Numbers.
Lower bound for the number of steps a 6-state, 5-tuple Turing machine can take (given an initially blank tape), before eventually halting. This machine, which was found by Pavel Kropitz in June 2010, effectively performs the iterated operation N → 3N+2 twice (from an initial N=0) before halting, a Collatz-like iteration. It leaves 3.515×1018267 marks in the tape, and is called a "busy beaver" because no other known 6-state machine produces more marks (except ones that run forever). It supplants the previous record belonging to a Ligoki machine, which ran for 2.5×102879 steps. See 107 and 47176870 for more.
The value of a myriad to the power of itself, written (by the system of Apollonius of Perga) as a script mu μ directly above a capital mu Μ. Significantly larger numbers were contemplated by Archimedes in The Sand Reckoner.
In 1989, Chen and Wang improved on Vinogradov's 1937 result (see 3.248...×106846168) showing that the weak Goldbach conjecture is true for all numbers larger than this. (Some sources give ee11.503 ≈ 3.33×1043000.) The result was later improved again, see e3100.
(2007 twin prime record)
An interesting, but not particularly useful theorem by Clement in 1949 states that n and n+2 are twin primes if and only if 4(n-1)! + n + 4 is divisible by n(n+2). The reason this is not particularly useful is because of the size of the factorial. For n=1.415..×1058710, 4(n-1)! is about 108.3116888×1058714.
1069882+3×1034941+1 is a large palindromic prime discovered by Daniel Heuer in 2002. As you can tell from the formula, its digits consist of a 1 followed by (69882-34941-1)=34940 zeros, followed by an 3, followed by another 34940 zeros, and then a 1. Because it is prime, it can be used to construct a large Smith number.
In The Hitchhiker's Guide to the Galaxy by Douglas Adams, this number is used when stating the odds against Ford and Arthur being rescued by a passing spaceship just after being thrown out an airlock. (This number is from the radio programme; for the book and the TV programme it was changed to 2260199.) It is one of the largest numbers used in a work of fiction. The same part of the story mentions monkeys and Hamlet; see 1040000 and 3.196×10282303.
See also 42.
As of 2007, the largest known factorial prime, defined as any value N!-1 or N!+1 that is prime.
8.729665098×10200699 = 3756801695685×2666669±1
(2011 twin prime record)
As of late 2011, 3756801695685×2666669-1 and 3756801695685×2666669+1 were the largest known pair of twin primes.
(Archimedes' Cattle Problem)
The solution to the larger (restricted) form of Archimedes' Cattle Problem. The problem was stated roughly as follows:
If you are diligent and wise, O stranger, compute the number of cattle of the Sun, who once upon a time grazed on the fields of the Thrinacian isle of Sicily, divided into four herds of different colours one milk white, another glossy black, the third yellow, and the fourth dappled. [...] The number of white bulls was equal to (1/2+1/3) the number of black bulls plus the total number of yellow bulls. The number of black bulls was (1/4+1/5) the number of dappled bulls plus the total number of yellow bulls. The number of spotted bulls was (1/6+1/7) the number of white bulls, plus the total number of yellow bulls. The number of white cows was (1/3+1/4) the total number of the black herd. The number of black cows was (1/4+1/5) the total number of the dappled herd. The number of dappled cows was (1/5+1/6) the total number of the yellow herd. The number of yellow cows was (1/6+1/7) the total number of the white herd.
If you can accurately tell, O stranger, the total number of cattle of the Sun, including the number of cows and bulls in each colour, you would not be called unskilled or ignorant of numbers, but not yet shalt thou be numbered among the wise. But understand also these conditions: [The white bulls could stand together with the black bulls in rows, such that the number of cattle in each row was equal and that number was equal to the total number of rows, thus forming a perfect square. And the yellow bulls could stand together with the dappled bulls, with a single bull in the first row, two in the second row, and continuing steadily to complete a perfect triangle.] If thou art able, O stranger, to find out all these things and gather them together in your mind, giving all the relations, thou shalt depart crowned with glory and knowing that thou hast been adjudged perfect in this species of wisdom.
If you solve just the first part of the problem, the smallest solution for the total number of cattle is 50389082. But if you add the additional two constraints in the second part, the solution is much higher about 7.76×10202544. It took until 1880 to find this answer, published by Amthor.
In 1931, in a letter to the New York Times, it was written
Since it has been calculated that it would take the work of a thousand men for a thousand years to determine the complete [exact] number [of cattle], it is obvious that the world will never have a complete solution.
of course, digital computers made the exact calculation possible, and the number was first calculated in 1965 by Williams, German and Zarnke on an IBM 7040. The 202545-digit number was first published in 1981 by Nelson. In 1998, Vardi showed that the number was the value of
25194541/184119152 × (109931986732829734979866232821433543901088049 + 50549485234315033074477819735540408986340 √4729494) 4658
rounded up to the nearest integer. In 2001, Nygrén showed how the problem could be solved in a manner simple enough (perhaps) to be known to the ancients (although it would not have enabled them to actually calculate the value of the solution, just prove that there is a solution and show how to calculate it).
(Reference: Chris Rorres' pages on Archimedes)
See also See also 3121.
(monkey typing "Hamlet")
The odds against a monkey typing out Shakespeare's Hamlet entirely by chance, based on a 35-key typewriter and 182831 characters (including spaces) in Hamlet. See also 1040000 and 1.95×101834097. (Note: this value used to be listed under 6.8738×1041689 = 3527000 and attributed to Dave Renfro, but I could not verify the source and the value was clearly wrong, so I have deprecated the attribution and recalculated the value.)
This is the first Mersenne prime found by a participant in the GIMPS (Great Internet Mersenne Prime Search) project. Many much larger primes have been found by the same group, including the current record largest prime.
This is "Borge's number", the number of books in the Library of Babel described in his short story by that name. Each book has 410 pages, with 40 lines of 80 characters on each page; there are 25 possible characters, and there is a book for every possible combination of characters. Thus, the library contains every work of fiction, both good and bad, every true newspaper account and countless untrue accounts, a biography of everyone who has ever lived and everyone yet to be born. Of course, an overwhelmingly large fraction of the books are just filled with random meaningless sequences of characters. See also 1040000, 2.748×1080588, 3.196×10282303 and 103000000.
This was the record-holder for largest known prime when it was discovered in 1999. It is a Mersenne prime, and its status as largest known prime was later superceded by 213466917-1. The current record is here.
This is (approximately) the maximum value that can be represented in the floating-point format used by PARI/GP, the free open-source symbolic math package developed at University Bordeaux, France.
According to Crandall , the odds against a parrot, randomly typing on a keyboard, reproducing The Hound of the Baskervilles on its first attempt, are 103000000 to 1. See also 1040000, 2.748×1080588, 3.196×10282303, and 1.95×101834097.
This number has the somewhat contrived name "milli-millillion". It is similar to the name "milli-millions" for the 1000000th power of a million (by the "long scale", i.e. 106000000) given by W. D. Henkle in the Ohio Educational Monthly, 1860. In the modern "short scale" that name would be used for 103000003. The author was called "Professor Henkle" in 1904 by Brooks, who also changed the name of the number to "milli-millillions"; this was cited by Dmitri Borgmann in 1968; see this discussion. For a long time it was the largest example I had seen of a number name in the Latin-prefix system which includes the more official names billion, decillion, vigintillion, etc.; however Conway and Wechsler created a system that extends arbitrarily far, and there have also been many ad-hoc attempts.
Quick index: if you're looking for a specific number, start with whichever of these is closest: 0.065988... 1 1.618033... 3.141592... 4 12 16 21 24 29 39 46 52 64 68 89 107 137.03599... 158 231 256 365 616 714 1024 1729 4181 10080 45360 262144 1969920 73939133 4294967297 5×1011 1018 5.4×1027 1040 5.21...×1078 1.29...×10865 1040000 109152051 101036 101010100 footnotes Also, check out my large numbers and integer sequences pages.