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# Notable Properties of Specific Numbers

(centillion)

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 [155] gives a usage from 1852, almost 100 years before the invention of the name googolplex. See also vigintillion, millillion and milli-millillion.

1.797693134862×10308 ≈ 21024

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.

The prime counting function, Sloane's A0720, can be computed without actually finding all the primes in question. More on the MathWorld page85.

250150 ≈ 4.9×10359

The estimate of the number of games of Go cited in a 2016 report on Google DeepMind AlphaGo, the Go-playing program that successfully beat a human champion in 2016. See also 3580 and 4.63×10170.

(the te Riele revision of "Skewes' number")

This is the revised, smaller value of the Skewes number, equal to ee27/4. It was given by H. J. J. te Riele in 1987. See also 1.397162914×10316 and 1.53×101165.

(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.

10421

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

In the Knuth -yllion naming system, 10421 is ten myriad tryllion quintyllion sextyllion; in the more mainstream Conway-Wechsler system, it is ten noventrigintacentillion.

10500

According to [209], 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. Each such universe will have its own specific physical constants and other details concerning its laws of physics; this number describes the number of ways those things can vary, while the actual number of multiple universes being created would be a lot larger, e.g. 101077 or the various larger numbers listed here:

1.7989021000...×10571 = 20!×319 × 30!×229 × (60!/(5!12))6 × (60!)2 / 210

(Teraminx puzzle)

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.

10600

The value of centillion in the long scale system; see 10303.

10800

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. The reader's original question was "How many unique English tweets are possible?", so we are not contemplating limits of Internet bandwidth or server storage capacity, merely considering how many possibilities the Twitter user has to choose from.

At the time, Twitter's character-counting description did not mention CJK (Chinese-Japanese-Korean) glyphs nor the two character "weight" that CJK glyphs have now. Munroe stated "You have all of Unicode to play with, which has room for over a million different characters" just before giving the upper bound ("could be as high as 10800"). The 140th root of 10800 is about 500000, suggesting that Munroe used that as an estimate of the number of available Unicode characters and raised it to the 140th power. At the time (late February 2013) there were only 249,764 assigned "code points" (characters plus many non-character things) and more than half of those were codes in the "private use area" which would not be available for use in a tweet. Assuming the actual number of Unicode characters at that time (which is 110,117) and still counting each as 1, the estimate drops a bit to about 7.23×10705.

In the time since then Twitter has increased the limit on tweets to 280 characters and now counts CJK glyphs as 2 (most all of the other languages' alphabets count as 1, regardless of the number and type of diacriticals or combining marks). The counting for emoji, etc. is pretty complicated. Most importantly however, URLs are shortened to 23 characters using Twitter's t.co domain, and they can effectively be used to encode a lot more information. There is probably a limit, but it is unclear what it is — I'll just average my entire browser history since 2010 (1.2 million URLs over 12 years) which is 95 characters (in ASCII, counting the length as seen in the browser address bar, not the longer length used internally in the HTTP protocol that requires %-escaping per RFC 3986). Coincidentally 95 is also the number of printable ASCII characters. In a 280-character Tweet you can get 12 URLs (with 4 characters left over), so I would estimate the number of possible Tweets as (9595)12 = 4×102254.

If would seem that Shannon's theorem of information density is being violated, but this does not in fact happen because the 23-character size of a shortened URL reflects the fact that only a very tiny fraction of possible URLs have ever actually been included in a Tweet. We are not asking "How many Tweets would be possible if all possible Tweets actually got Tweeted and Twitter had to store all of them?" — we are asking "How many possibilities does a user have to choose from when they compose a Tweet?"

1.29149...×10865

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 457936006084633875­260691932542213506579481395376­080192442872707759996212114957­373537195900697943283211344130­969977204683723647091975242566­556807073476262370119366712949­612051508874565615465951982148­103948322515169952026557331614­199239782652240565877185274882­891122589783986489974588207230­026310073238799349251084594897­863556829085566422093207975001­895285824382289647389848615424­710629561529529589935914349946­023950287863307022313442880758­800532983282085207377266536998­146723331964258315488766981883­904240306133944424567760471103­539279962416731476757145320641­439420037963516042879919957607­890943287019373144639492683640­803862704805497501551907216898­677744138585826270309663329962­841518933729157858558919253022­063551926057138672786596389094­200184031909805595086778342937­081605771699885426749776777391­919555685119629369584896777148­250878775274042686107865894781­763500774758450843791837394393­056896301600021929961984000000. The factorisation 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.

2×10963

Approximate value of the "lexicographically last" prime number, in American style English if Conway-Wechsler naming is used but without the post-103000 "-illi-" extension and without ever using "and". The name of the prime would start "two vigintitrecentillion two vigintillion two undecillion ..."; see 2×1063 For reasons explained in the entry for 2.1359...×1096, there are almost certainly larger, lexicographically later primes past 103000. However as the numbers get bigger it takes more time and effort to determine if they are prime, so there would always be new winners for such a contest.

7.7263039555...×10992 = 20!×319 × 30!×229 × (60!/(5!12))12 × (60!)3 / 217

(Petaminx puzzle)

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.

9.9999999...×10999

Some more expensive pocket calculators (such as the TI-85 and TI-92) max out at 9.9999999...×10999. See also 9.9999999...×1099 and the computer overflow values starting with 3.4028236692093×1038.

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.

1.111111...×101031

The largest known repunit prime consists of 1032 1's in a row. It is a component of several large Smith numbers: see 1010694986, 1013614514, and 1032066910.

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 was the largest N×N×N cube ever made as a real physical object (a record broken in 2017, see 1.869666101×104099). It has 1539 parts and at least two copies have been made via the 3D printing service shapeways costing well over \$10,000 US. Videos of it can be seen here (inventor Oskar van Deventer) and here (cubist/reviewer RedKB).

The number of combinations is a 1055-digit number, and is about 67 quinquagintatrecentillion (using the Conway-Wechsler naming system, "quinseptuagintacentilliard" using the long scale). In the RedKB video this number is pronounced "quin-quagin-tat-reken-tillion", but it's spelled the same way. The terms of the formula are computed similarly to the description in the entry for 7×7×7. In the video one can see all the digits scroll by, which for the record are: 66,­909,­260,­871,­052,­009,­626,­140,­831,­457,­599,­196,­711,­140,­812,­269,­154,­070,­729,­060,­136,­529,­449,­625,­780,­211,­961,­895,­693,­820,­570,­513,­604,­163,­602,­868,­942,­801,­633,­627,­363,­413,­148,­772,­664,­738,­570,­971,­988,­412,­147,­490,­850,­469,­267,­091,­069,­898,­537,­146,­037,­768,­890,­069,­934,­919,­884,­249,­763,­818,­629,­080,­668,­367,­898,­685,­033,­459,­370,­133,­844,­075,­322,­446,­474,­048,­403,­397,­592,­421,­266,­564,­641,­031,­053,­781,­182,­835,­951,­043,­902,­666,­703,­934,­718,­275,­733,­629,­773,­072,­428,­119,­603,­386,­280,­810,­232,­743,­294,­106,­725,­017,­906,­015,­726,­602,­505,­404,­809,­355,­600,­713,­515,­400,­760,­343,­408,­510,­054,­774,­806,­467,­063,­695,­824,­637,­124,­911,­945,­446,­317,­465,­833,­055,­520,­836,­975,­861,­238,­244,­940,­397,­333,­234,­336,­971,­270,­687,­092,­383,­804,­133,­631,­886,­114,­309,­853,­819,­332,­336,­282,­986,­834,­777,­948,­178,­464,­656,­888,­802,­372,­250,­927,­074,­981,­140,­246,­608,­824,­577,­036,­094,­710,­201,­099,­095,­240,­641,­256,­513,­217,­598,­802,­423,­874,­027,­822,­421,­584,­587,­650,­039,­125,­516,­202,­912,­205,­481,­540,­427,­864,­199,­947,­576,­722,­221,­866,­866,­102,­507,­350,­876,­922,­115,­628,­881,­880,­203,­115,­212,­216,­766,­503,­665,­426,­445,­956,­786,­264,­399,­133,­302,­962,­649,­600,­884,­736,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000.

(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.

1.9079700752...×101280 = 24253-1

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?"

2.8554254223...×101331 = 24423-1

M4423, the 20th Mersenne Prime; see 1.9079700752...×101280.

e31002.05...×101346

In 2002, M.-Ch. Liu and T. Wang improved on Chen and Wang's 1989 result on the weak Goldbach conjecture (1043000), showing that the conjecture is true for all numbers larger than this. See 8.005792×104008659. (Some sources give "2×101346" rather than "e3100".)

8.88888888888888...×101349

The "Trinity Hall Prime", a prime number whose digits can be arranged into a rectangle containing an image of the coat of arms of Trinity Hall, the oldest college of Cambridge University.

888888888888888888888888888888 888888888888888888888888888888 888888888888888888888888888888 888111111111111111111111111888 888111111111111111111111111888 888111111811111111118111111888 888111118811111111118811111888 888111188811111111118881111888 888111188811111111118881111888 888111888811111111118888111888 888111888881111111188888111888 888111888888111111888888111888 888111888888888888888888111888 888111888888888888888888111888 888111888888888888888888111888 888811188888888888888881118888 188811188888888888888881118881 188881118888888888888811188881 118888111888888888888111888811 111888811118888888811118888111 111188881111111111111188881111 111118888111111111111888811111 111111888811111111118888111111 111111188881111111188881111111 111111118888811118888811111111 111111111888881188888111111111 111111111118888888811111111111 111111111111888888111111111111 111111111111118811111111111111 111111111111111111111111111111 062100000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000001 is a prime, with 1350 digits. JF McKee

Trinity Hall was founded in the year 1350, and this prime has 1350 digits. It was found by James McKee and left as a parting gift to the college in 1994. Is is shown and discussed in a Numberphile video, "The Trinity Hall Prime".

101365

This is one "quattuorquinquagintaquadringentillion" using the Conway-Wechsler system for naming powers of 1000 from 1036 on upwards (the smaller ones, up to 1033 decillion, are widely used and in most dictionaries. In particular, they name every power of 1000 from undecillion to 103000 novenonagintanongentillion with a single word (without hyphens). Of these, the longest is quattuorquinquagintaquadringentillion with 37 letters. See also 10763365

2.215011...×101428

See 6.700591...×1074.

4.6×101439

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.

3.002327716...×101730

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.

2.518504×101906

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.

4×102254

Estimate of the number of possible choices when composing a Tweet, assuming one fills the Tweet with URLs and takes advantage of Twitter's policy of shortening each URL to 23 characters. This is based on an average URL length of 95 (from a ton of browser history) and Tweet length of 280 characters; see 10800 for more.

2.5×102879

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.

(millillion)

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.

7.232325232...×103119

72323252323272325252 × (103120-1) / (1020-1) + 1 is a huge prime number whose digits are all prime: 156 copies of 72323252323272325252, with the final digit changed to a 3. It was reported by Paulo Ribenboim in his 2000 book "My Numbers, My Friends". See also 2357 and 7.777777777...×1099..

1.159267176...×104094 = 24!15 × (24!/246)((312-1)/4) × 12! × 28 × 37

Incorrect value of 1.8696661...×104099 (permutations of an order-33 Rubik's cube).

104096

The value of the number called zài in one ancient Chinese system for naming large numbers36. In this system, The successive names , zhào, etc. name successive squares of wàn (which is 104), thus =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.

1.8696661...×104099 = 24!15 × (24!/246)((312-1)/4) × 12! × 211 × 8! × 37 / 2

The number of ways to arrange a 33×33×33 Rubik's cube. As of 2017, this was the highest-order N×N×N cube ever made as a real physical object. Mechanical engineer and puzzle designer GrĂ©goire Pfennig created and 3-D printed 6153 parts (6146 of which are visible) to create a puzzle about 30 cm on a side.

(In the video on his channel "Greg's Puzzles" it is claimed the puzzle has 1.159267176...×104094 combinations, which is less than reality by a ratio of 161280 = 7!×25 = 8!×4.)

1.000...×104180 = 102085+1×(818181818+102085+9)+1

This is a 4181-digit number, and is prime, and looks like:

1000000000000...0081818181800...0000000000001

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)

1.0000.....×104594

104594+3×102297+1 is a large palindromic prime discovered[172] 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.

3.98...×106020 = 220000

This is 220000, my estimate of the number of viable and distinguishable results that could be achieved by mutations (or deliberate alteration) of the base-pairs in the human genome. There are about 19000 to 20000 protein-coding genes in the human DNA (see Human genome), but for each gene, at most a few individual amino acids could be changed, and only to one or two other alternatives (if more than that are changed, the entire protein would fail, and if multiple proteins are altered in such a destructive manner, the combination would be fatal). I imagine that the total number of combinations that can survive and which would be distinguishable (by phenotype) from each other is about 2 to the power of the number of coded proteins, or 220000. See also 8.6×104515449 and 3.015...×103576838408.

Contemplating this number brings up ethical issues (such as eugenics) and uncertainty in definitions (there are viable embryos in the early days after fertilisation that do not survive weeks later.

This topic came to me from a reader inspired by the opening of Richard Dawkins' book Unweaving the Rainbow: "[...] The potential people who could have been here in my place but who will in fact never see the light of day outnumber the sand grains of Arabia. Certainly those unborn ghosts include greater poets than Keats, scientists greater than Newton. We know this because the set of possible people allowed by our DNA so massively outnumbers the set of actual people. [...]" Notwithstanding variations in DNA, it is almost universally agreed that Keats and Newton are who they are mainly because of things that happened after their DNA was decided.

(Lucas-Lehmer test)

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.

1.0000...×106572

106572+3×103286+1 is a large palindromic prime discovered[172] 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.

2.703176857×106940

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.

Prior to the 50- and 75-move rules, the only limit on the length of a chess game was threefold repetition, producing the far larger Hardy's estimate.

(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).

For a while this was the largest known Leyland prime. It was later superceded by 7.00558×1025049 and 8.37300×1030007.

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.

3.515×1018267

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, and 1010106.935×1018705352.

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:

 m n=0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 8 ... S(n+3) - 3 where S(x) is the successor function S(x)=x+1 1 2 3 4 5 6 7 8 9 ... 2+(n+3) - 3 2 3 5 7 9 11 13 15 17 ... 2(n+3) - 3 3 5 13 29 61 125 253 509 1021 ... 2n+3 - 3 4 13 65533 265536-3 2265536-3-3 ...

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 OEIS sequence A51254 and is related to the problem of finding and proving the value of Mills' constant.

2.959364...×1021077 = 235235×270000-1

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.

7.00558031...×1025049 = 67535122+51226753

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.

See 593 and 5.19344×1015070.

1.0000...×1028572

1028572+8×1014286+1 is a large palindromic prime discovered[172] 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 [172] to construct a record Smith number.

1.948828382...×1029603 = 832780

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.

8.37300356...×1030007 = 86562929+29298656

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.

7.412000003×1036534

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.

1040000 = 1000010000

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.

This number is also cited by Knuth as "the number of trials" before a monkey sitting at a typewriter would produce the text of Hamlet65; see 3.196×10282303.

1043000

In 1989, Chen and Wang improved on Vinogradov's 1937 result (see 8.005792×104008659) 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.

2.0014732742×1051089 = 33218925×2169690-1

For a while, 33218925×2169690-1 and 33218925×2169690+1 were the largest known pair of twin primes. They have since been surpassed by 2003663613×2195000±1

1.41572626..×1058710 = 2003663613×2195000-1

(2007 twin prime record)

As of early 2007, 2003663613×2195000-1 and 2003663613×2195000+1 were the largest known pair of twin primes. They were since surpassed by 3756801695685×2666669±1.

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.

1.0000...×1069882

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.

5.05640407...×1078327 = 2260199

2.74858523...×1080588 = 2267709

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.

5.81257947...×10142890 = 34790! - 1

As of 2007, the largest known factorial prime, defined as any value N!-1 or N!+1 that is prime.

5.086520...×10189682

See 6.700591...×1074.

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 smallest solution is much higher — about 7.76×10206544. 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)

10272000

A recent (2022) estimate of "the number of universes made possible by string theory", described in this video by Numberphile. From the beginning, string theory has been thought to allow many possible universes, where "universe" represents a plausible way to handle the extra six dimensions (of the ten in string theory). Generally this idea is described as if the six extra dimensions are very small multidimensional manifolds (like a torus, the surface of which is a two-dimensional manifold), smaller than the limits of what we can measure. In earlier times it was thought there were about 10500 possible configurations of the multidimensional manifolds ([209], [239]), but as the "string theory landscape" has evolved, more configurations become possible. This larger number 10272000 comes from [234], which states it most tersely, "<*"Applying the Ashok-Denef-Douglas estimation method to elliptic Calabi-Yau fourfolds suggests that a single elliptic fourfold Mmax gives rise to O(10272,000) F-theory flux vacua.".

(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.)

8.147175681×10420920 = 21398269-1

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.

10763365

Using the Conway-Wechsler system for naming powers of 1000, this number has the longest name of all such numbers less than 101000000: quattuorquinquagintaducentilliquattuorquinquagintaquadringentillion, with 67 letters. See also 101365.

2.0650635..×101262611 = 2222-1

A Fermat number, which has been proven composite without determining any factors. See 1016.

1.9560399...×101834097 = 25^(410×40×80)

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.

4.370757...×102098959 = 26972593 - 1

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.

4.26448742×102525222 = 2223

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.

103000000 = 103×106

According to Crandall [166], 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.

(milli-millillion, milliamilliatillion)

This number was a long-term record-holder for the largest Chuquet-like number name in an organised system by an individual person, i.e. ad-hoc Chuquet extension. 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 presented short scale definitions and changed this name from "milli-millions" to "milli-millillions"; this was cited by Dmitri Borgmann in 1968[140]; see this discussion. For a long time it was the best-known and largest example of a number name in a Latin-prefix system which includes the more official names billion, decillion, vigintillion, etc..

The current "majority favourite" Chuquet-like system is that of Conway and Wechsler, published in a 1986 book [152]. It is by far the best-researched Chuquet-like system and extends arbitrarily far. In their system 103000003 is "milliamilliatillion".

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.

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