Notable Properties of Specific Numbers
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(centillion)
The number centillion, which for a long time was the largest number with a singleword name in English (and many other languages that use the Chuquet names). In the majority of the Englishspeaking world centillion meant 10^{600}. 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 millimillillion.
1.797693134862×10^{308} ≈ 2^{1024}
This is (approximately) the maximum value that can be represented in the commonlyused IEEE 754 doubleprecision (1+11+52 bit) floatingpoint format.
See also 3.4028236692093×10^{38}, 1.1897314953572318×10^{4932}, 4.26448742×10^{2525222} and 1.4403971939817846×10^{323228010}.
(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 e^{ee79} ≈ 10^{101034}, the "Skewes number".
The approximation 1.39822×10^{316} 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 1.397162914×10^{316}.
The prime counting function, Sloane's A0720, can be computed without actually finding all the primes in question. More on the MathWorld page^{85}.
See also 906150257.
The estimate of the number of games of Go cited in a 2016 report on Google DeepMind AlphaGo, the Goplaying program that successfully beat a human champion in 2016. See also 35^{80} and 4.63×10^{170}.
8.1847946207224960623437×10^{370}
(the te Riele revision of "Skewes' number")
This is the revised, smaller value of the Skewes number, equal to e^{e27/4}. It was given by H. J. J. te Riele in 1987. See also 1.397162914×10^{316} and 1.53×10^{1165}.
(the "singleperturbation 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 1.41×10^{408} distinct choices (as limited by the uncertainty principle). The formula for this is equivalent to the universe's 4dimensional 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 singleperturbation 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 spacetime on large scales, which throws off any such calculation quite a bit.
See also 10^{1016}, 10^{1.877×1054}, 10^{1077}, 10^{1082}, 10^{10166}, 10^{3.79×10281}, 10^{10375}, 10^{5.7×10410}, 10^{1010000000}, 10^{1010122}, and 10^{101.51×103883775501690}.
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 10^{7}. He gives names for powers of ten up to the tallaksana, 10^{53}. He then describes successive "numerations", the dvajagravati=10^{99}, the dvajagranisamani=10^{145}, and several more culminating in 10^{421}, which is given the name uttaraparamanurajahpravesa^{28}. 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 placevalue notation with a symbol for zero.^{29}
In the Knuth yllion naming system, 10^{421} is ten myriad tryllion quintyllion sextyllion; in the more mainstream ConwayWechsler system, it is ten noventrigintacentillion.
See also 10^{140} and 10^{3.7218×1037}.
According to [208], 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. 10^{1077} or the various larger numbers listed here:
See also 10^{40}, 10^{272000}, 10^{1016}, 10^{1.877×1054}, 10^{1077}, 10^{1082}, 10^{10166}, 10^{3.79×10281}, 10^{10375}, 10^{5.7×10410}, 10^{1010000000}, 10^{1010122}, and 10^{101.51×103883775501690}.
1.7989021000...×10^{571} = 20!×3^{19} × 30!×2^{29} × (60!/(5!^{12}))^{6} × (60!)^{2} / 2^{10}
(Teraminx puzzle)
This is the number of ways to arrange the pieces on a Teraminx, a dodecahedronshaped 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 CADCAM prototyping machines. Here is a video of one being assembled, and here you can see the puzzle being manipulated.
See also 1.0067×10^{68}, 9.1197×10^{262}, and 7.7263×10^{992}.
The value of centillion in the long scale system; see 10^{303}.
Randall Munroe made this estimate of the number of possible 140character 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 charactercounting description did not mention CJK (ChineseJapaneseKorean) 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 10^{800}"). The 140^{th} root of 10^{800} is about 500000, suggesting that Munroe used that as an estimate of the number of available Unicode characters and raised it to the 140^{th} power. At the time (late February 2013) there were only 249,764 assigned "code points" (characters plus many noncharacter 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 7.23×10^{705}.
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 280character Tweet you can get 12 URLs (with 4 characters left over), so I would estimate the number of possible Tweets as (95^{95})^{12} = 4×10^{2254}.
If would seem that Shannon's theorem of information density is being violated, but this does not in fact happen because the 23character 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?"
See also 2.28×10^{46} and 2.45995..×10^{200}.
A 6state turing machine, found by Heiner Marxen and Juergen Buntrock in 2001 March, takes 3.00233×10^{1730} steps before halting with 1.29149×10^{865} 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'=2^{K×X} several times in a row, with a chaotic deterministic lowprobability exit condition. Its record was broken by Terry and Shawn Ligocki, whose machine leaves 4.6×10^{1439} marks.
4.57936...×10^{917} = 2^{15}×3^{10}×5^{6}×7^{5}×11^{4}×...×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 10^{3535}, was found by Achim Flammenkamp^{7}. See also 12, 840, 45360, 720720, 3603600, 245044800, 278914005382139703576000 and 2054221614063184107682218077003539824552559296000.
Approximate value of the "lexicographically last" prime number, in American style English if ConwayWechsler naming is used but without the post10^{3000} "illi" extension and without ever using "and". The name of the prime would start "two vigintitrecentillion two vigintillion two undecillion ..."; see 2×10^{63} For reasons explained in the entry for 2.1359...×10^{96}, there are almost certainly larger, lexicographically later primes past 10^{3000}. 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...×10^{992} = 20!×3^{19} × 30!×2^{29} × (60!/(5!^{12}))^{12} × (60!)^{3} / 2^{17}
(Petaminx puzzle)
This is the number of ways to arrange the pieces on a Petaminx, a dodecahedronshaped 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.
See also 1.0067×10^{68}, 9.1197×10^{262}, and 1.7989×10^{571}.
Some more expensive pocket calculators (such as the TI85 and TI92) max out at 9.9999999...×10^{999}. See also 9.9999999...×10^{99} and the computer overflow values starting with 3.4028236692093×10^{38}.
1.97231222789×10^{1015} = 2^{172}3^{9}5^{11}7^{5}11^{11}13^{172}17^{11}19^{1}23^{13}29^{13}31^{7}37^{172}41^{11}43^{1}47^{13}53^{13}
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 a_{2} ∀ (~(a_{2}(0)) ∨ a_{2}(0)). This formula has 16 basic signs, with individual Gödel numbers 17^{2}, 9, 11, 5, 11, 17^{2}, 11, 1, 13, 13, 7, 17^{2}, 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.
The largest known repunit prime consists of 1032 1's in a row. It is a component of several large Smith numbers: see 10^{10694986}, 10^{13614514}, and 10^{32066910}.
6.690926087×10^{1054} = 2^{8} × (8!/2)×3^{7} × (12!/2)×2^{11} × (24!/2)^{63} / ((24^{6})/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 1.869666101×10^{4099}). 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 1055digit number, and is about 67 quinquagintatrecentillion (using the ConwayWechsler naming system, "quinseptuagintacentilliard" using the long scale). In the RedKB video this number is pronounced "quinquagintatrekentillion", 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.
See also 3674160, 4.3252×10^{19}, 7.4012×10^{45}, 1.9501×10^{160}, 1.417×10^{277}, and 1.8696×10^{4099}.
(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 1.53×10^{1165} and 1.65×10^{1165}. This was the first published computed estimate, and improved significantly on Skewes' estimate 10^{101034}. See also 1.397×10^{316}.
1.9079700752...×10^{1280} = 2^{4253}1
M_{4253}, 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 M_{4423}. 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...×10^{1331} = 2^{4423}1
M_{4423}, the 20th Mersenne Prime; see 1.9079700752...×10^{1280}.
In 2002, M.Ch. Liu and T. Wang improved on Chen and Wang's 1989 result on the weak Goldbach conjecture (10^{43000}), showing that the conjecture is true for all numbers larger than this. See 8.005792×10^{4008659}. (Some sources give "2×10^{1346}" rather than "e^{3100}".)
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".
This is one "quattuorquinquagintaquadringentillion" using the ConwayWechsler system for naming powers of 1000 from 10^{36} on upwards (the smaller ones, up to 10^{33} decillion, are widely used and in most dictionaries. In particular, they name every power of 1000 from undecillion to 10^{3000} novenonagintanongentillion with a single word (without hyphens). Of these, the longest is quattuorquinquagintaquadringentillion with 37 letters. See also 10^{763365}
A former recordholder for the 6state busy beaver Turing machine takes about 2.5×10^{2879} steps before halting with 4.6×10^{1439} ones on the tape. The machine was discovered by Terry and Shawn Ligocki in 2007, and overtook a MarxenBuntrock machine that left 1.3×10^{865} marks. The machine that surpassed it leaves 3.515×10^{18267} marks. See also 107 and 47176870.
Number of steps taken by a certain 6state, 5tuple Turing machine before halting. It was a recordholder 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 2.5×10^{2879}. See 107 for more.
An "11fold multiperfect number" found by Woltman on 2001 Mar 13^{th}. The sum of its divisors (including 1 and itself) is exactly 11 times its value. See also 120, 496, 30240, and 154345556085770649600.
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 10^{800} for more.
Lower bound for the number of steps a 6state, 5tuple 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 MarxenBuntrock machine, which took 3×10^{1730} 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.
See also 10^{3000003}.
72323252323272325252 × (10^{3120}1) / (10^{20}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...×10^{99}..
1.159267176...×10^{4094} = 24!^{15} × (24!/24^{6})^{((3121)/4)} × 12! × 2^{8} × 3^{7}
Incorrect value of 1.8696661...×10^{4099} (permutations of an order33 Rubik's cube).
The value of the number called zài in one ancient Chinese system for naming large numbers^{36}. In this system, The successive names yì, zhào, etc. name successive squares of wàn (which is 10^{4}), thus yì=10^{8}, zhào=10^{16}, and so on up to zài=10^{4096}. In modern usage, zài is "merely" 10^{44}. In the Knuth yllion naming system, 10^{4096} is one decyllion; in the more mainstream ConwayWechsler system, it is ten milliquattuorsexagintatrecentillion.
1.8696661...×10^{4099} = 24!^{15} × (24!/24^{6})^{((3121)/4)} × 12! × 2^{11} × 8! × 3^{7} / 2
The number of ways to arrange a 33×33×33 Rubik's cube. As of 2017, this was the highestorder N×N×N cube ever made as a real physical object. Mechanical engineer and puzzle designer GrĂ©goire Pfennig created and 3D 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...×10^{4094} combinations, which is less than reality by a ratio of 161280 = 7!×2^{5} = 8!×4.)
See also 6.69092...×10^{1054}.
1.000...×10^{4180} = 10^{2085+1}×(818181818+10^{2085+9})+1
This is a 4181digit 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)
See also 1.000...×10^{101}.
10^{4594}+3×10^{2297}+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 (459422971)=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.
See also 1.000...×10^{6572}, 1.000...×10^{28572}, and 1.000...×10^{69882}.
1.1897314953572318×10^{4932} ≈ 2^{16384} = 2^{214}
This is (approximately) the maximum value that can be represented in several implementations of IEEE 754 extended double floatingpoint formats, and the IEEE 754r "binary128" format. They all have a 15bit exponent field. In most other respects, the various extended double formats differ. The most common is exemplified by the Intel IA64 architecture's 10byte "extended doubleprecision" which has a 63bit mantissa; less common is the 16byte "quadruple precision" (such as found on Digital VAX and Alpha systems) with a 112bit mantissa. The IEEE 754 specifications for "extended" formats allow the implementer to choose pretty nearly any exponent and mantissa size they want.
See also 3.4028236692093×10^{38}, 1.797693134862×10^{308}, 1.948828382..×10^{29603}, 4.26448742×10^{2525222} and 1.4403971939817846×10^{323228010}.
(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 5000digit number, using the ECPP (Elliptic Curve Primality Proving) method^{46}. See also 10^{15000} and 1.7505×10^{20561}.
This is 2^{20000}, my estimate of the number of viable and distinguishable results that could be achieved by mutations (or deliberate alteration) of the basepairs in the human genome. There are about 19000 to 20000 proteincoding 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 2^{20000}. See also 8.6×10^{4515449} and 3.015...×10^{3576838408}.
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.
4.486791...×10^{6532} = 2^{21701}  1
(LucasLehmer test)
This is one of many Mersenne primes discovered by computer using the LucasLehmer test. The LucasLehmer test states that you can test a Mersenne number M_{n} for primeness by computing the sequence S_{1} = 4, S_{n+1} = S_{n}^{2}  2, and checking if S_{n1} divides evenly into M_{n}. If it does, M_{n} 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 2^{n}). The result is that today, nearly anyone with a home computer now has a shot at discovering Mersenne primes. In 1978, two highschool students (Noll & Nickel) discovered 2^{21701}1 on a local university mainframe computer, and by the late 1990's all new Mersenne primes were being discovered by individual personal computers.
10^{6572}+3×10^{3286}+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 (657232861)=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.
See also 1.000...×10^{4594}, 1.000...×10^{28572}, and 1.000...×10^{69882}.
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 26^{8848}, 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 10^{120} is much more realistic.
Prior to the 50 and 75move rules, the only limit on the length of a chess game was threefold repetition, producing the far larger Hardy's estimate.
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 10^{5000}.
5.19344195...×10^{15070} = 2638^{4405} + 4405^{2638}
For a while this was the largest known Leyland prime. It was later superceded by 7.00558×10^{25049} and 8.37300×10^{30007}.
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...×10^{15599} = (2255737522 × R_{15600}) / 1111111111 + 1
In 2002 Harvey Dubner and David Broadhurst^{40} 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. R_{15600} is the repunit with 15600 digits; note that R_{15600}/R_{10} = 100000000010000000001000...00001, a 15591digit 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 × R_{15600}) / 1111111111 + 1.
As of June 2010, the record for the 6state busy beaver Turing machine takes about 7.412×10^{36534} steps before halting with 3.515×10^{18267} ones on the tape. The machine was discovered by Pavel Kropitz in June 2010, and overtook a Ligocki machine that left 4.6×10^{1439} marks. See also 107 and 47176870, and 10^{10106.935×1018705352}.
2.0035299...×10^{19728} = 2^{65536}3
The Ackermann function
The Ackermann function is a function that grows very fast, but has a surprisingly innocentlooking definition. Using the twoargument version of Peters, A(m,n) = n+1 (if m=0) or A(m1,1) (if m>0 and n=0):> or A(m1,A(m,n1)) <$:(for remaining cases). This function produces the following table:

The first row is the positive integers, and each subsequent row is an Nthterm 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...×10^{20561} = (((((((((2^{3}+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 generalpurpose 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 2.4GHz Opteron running for over 6 years^{46}. The sequence: 2, 2^{3}+3, (2^{3}+3)^{3}+30, ... is OEIS sequence A51254 and is related to the problem of finding and proving the value of Mills' constant.
See also 10^{5000}.
2.959364...×10^{21077} = 235235×2^{70000}1
The "largest known easytoremember prime", discovered by the "Amdahl six", a team of large prime hunters. They discovered this 21078digit prime number as part of a larger project to identify large primes fitting the pattern p = A 2^{B} +/ 1. It can be remembered by its formula 235235 × 2^{70000}  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...×10^{19}.
7.00558031...×10^{25049} = 6753^{5122}+5122^{6753}
For a while this was the largest known prime number of the form x^{y} + y^{x} (where x and y are integers greater than 1), i.e. a prime that is also a Leyland number. It was later superceded by 8.37300×10^{30007}.
See 593 and 5.19344×10^{15070}.
10^{28572}+8×10^{14286}+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 (28572142861)=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.
See also 1.000...×10^{4594} and 1.000...×10^{6572}, and 1.000...×10^{69882}.
1.948828382...×10^{29603} = 8^{32780}
This is (approximately) the maximum value that can be represented in the doubleprecision format on the Burroughs 6x00 family of mainframe computers, and is the highest overflow value for any hardware floatingpoint format I have heard of to date. Numerous softwareimplemented formats exceed it.
See also 3.4028236692093×10^{38}, 1.797693134862×10^{308}, 1.1897314953572318×10^{4932}, 4.26448742×10^{2525222} and 1.4403971939817846×10^{323228010}.
8.37300356...×10^{30007} = 8656^{2929}+2929^{8656}
As of late 2014, this is the largest known prime number of the form x^{y} + y^{x} (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 x^{y} + y^{x} (without the need to be prime) is called a Leyland number. Numberphile has a video about it here: Leyland Numbers.
See also 5.19344×10^{15070} and 7.00558×10^{25049}.
Lower bound for the number of steps a 6state, 5tuple 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 → 3^{N+2} twice (from an initial N=0) before halting, a Collatzlike iteration. It leaves 3.515×10^{18267} marks in the tape, and is called a "busy beaver" because no other known 6state machine produces more marks (except ones that run forever). It supplants the previous record belonging to a Ligoki machine, which ran for 2.5×10^{2879} 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.
This number is also cited by Knuth as "the number of trials" before a monkey sitting at a typewriter would produce the text of Hamlet^{65}; see 3.196×10^{282303}.
In 1989, Chen and Wang improved on Vinogradov's 1937 result (see 8.005792×10^{4008659}) showing that the weak Goldbach conjecture is true for all numbers larger than this. (Some sources give e^{e11.503} ≈ 3.33×10^{43000}.) The result was later improved again, see e^{3100}.
2.0014732742×10^{51089} = 33218925×2^{169690}1
For a while, 33218925×2^{169690}1 and 33218925×2^{169690}+1 were the largest known pair of twin primes. They have since been surpassed by 2003663613×2^{195000}±1
1.41572626..×10^{58710} = 2003663613×2^{195000}1
(2007 twin prime record)
As of early 2007, 2003663613×2^{195000}1 and 2003663613×2^{195000}+1 were the largest known pair of twin primes. They were since surpassed by 3756801695685×2^{666669}±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(n1)! + 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..×10^{58710}, 4(n1)! is about 10^{8.3116888×1058714}.
10^{69882}+3×10^{34941}+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 (69882349411)=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.
See also 1.000...×10^{4594}, 1.000...×10^{6572}, and 1.000...×10^{28572}.
5.05640407...×10^{78327} = 2^{260199}
2.74858523...×10^{80588} = 2^{267709}
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 2^{260199}.) It is one of the largest numbers used in a work of fiction. The same part of the story mentions monkeys and Hamlet; see 10^{40000} and 3.196×10^{282303}.
See also 42.
5.81257947...×10^{142890} = 34790!  1
As of 2007, the largest known factorial prime, defined as any value N!1 or N!+1 that is prime.
8.729665098×10^{200699} = 3756801695685×2^{666669}±1
(2011 twin prime record)
As of late 2011, 3756801695685×2^{666669}1 and 3756801695685×2^{666669}+1 were the largest known pair of twin primes.
7.760271406486818269530232833213...×10^{206544}
(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 7.76×10^{206544}. 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 202545digit 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 3121.
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 twodimensional manifold), smaller than the limits of what we can measure. In earlier times it was thought there were about 10^{500} possible configurations of the multidimensional manifolds ([208], [238]), but as the "string theory landscape" has evolved, more configurations become possible. This larger number 10^{272000} comes from [233], which states it most tersely, "<*"Applying the AshokDenefDouglas estimation method to elliptic CalabiYau fourfolds suggests that a single elliptic fourfold M_{max} gives rise to O(10^{272,000}) Ftheory flux vacua.".
3.196×10^{282303} = 35^{182831}
(monkey typing "Hamlet")
The odds against a monkey typing out Shakespeare's Hamlet entirely by chance, based on a 35key typewriter and 182831 characters (including spaces) in Hamlet. See also 10^{40000} and 1.95×10^{1834097}. (Note: this value used to be listed under 6.8738×10^{41689} = 35^{27000} 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.)
8.147175681×10^{420920} = 2^{1398269}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.
Using the ConwayWechsler system for naming powers of 1000, this number has the longest name of all such numbers less than 10^{1000000}: quattuorquinquagintaducentilliquattuorquinquagintaquadringentillion, with 67 letters. See also 10^{1365}.
2.0650635..×10^{1262611} = 2^{222}1
A Fermat number, which has been proven composite without determining any factors. See 10^{16}.
1.9560399...×10^{1834097} = 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 10^{40000}, 2.748×10^{80588}, 3.196×10^{282303} and 10^{3000000}.
4.370757...×10^{2098959} = 2^{6972593}  1
This was the recordholder 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 2^{13466917}1. The current record is here.
4.26448742×10^{2525222} = 2^{223}
This is (approximately) the maximum value that can be represented in the floatingpoint format used by PARI/GP, the free opensource symbolic math package developed at University Bordeaux, France.
See also 3.4028236692093×10^{38}, 10^{300}, 1.797693134862×10^{308}, 1.1897314953572318×10^{4932} and 1.4403971939817846×10^{323228010}.
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 10^{3000000} to 1. See also 10^{40000}, 2.748×10^{80588}, 3.196×10^{282303}, and 1.95×10^{1834097}.
(millimillillion, milliamilliatillion)
This number was a longterm recordholder for the largest Chuquetlike number name in an organised system by an individual person, i.e. adhoc Chuquet extension. The somewhat contrived name "millimillillion". It is similar to the name "millimillions" for the 1000000^{th} power of a million (by the "long scale", i.e. 10^{6000000}) given by W. D. Henkle in the Ohio Educational Monthly, 1860. In the modern "short scale" that name would be used for 10^{3000003}. The author was called "Professor Henkle" in 1904 by Brooks, who presented short scale definitions and changed this name from "millimillions" to "millimillillions"; this was cited by Dmitri Borgmann in 1968[140]; see this discussion. For a long time it was the bestknown and largest example of a number name in a Latinprefix system which includes the more official names billion, decillion, vigintillion, etc..
The current "majority favourite" Chuquetlike system is that of Conway and Wechsler, published in a 1986 book [152]. It is by far the bestresearched Chuquetlike system and extends arbitrarily far. In their system 10^{3000003} is "milliamilliatillion".
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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 5×10^{11} 10^{18} 5.4×10^{27} 10^{40} 5.21...×10^{78} 1.29...×10^{865} 10^{40000} 10^{9152051} 10^{1036} 10^{1010100} — — footnotes Also, check out my large numbers and integer sequences pages.
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