Notable Properties of Specific Numbers (page 19) at MROB

First page . . . Back to page 18 . . . Forward to page 20 . . . Last page (page 25) 1.786266437(26)×1041 This is 2 to the power of the reciprocal fine-structure constant 137.0359... using the CODATA 2022 recommended value of the latter. It is a simple use of the popular fine structure constant to produce a value close to the Dirac ratio 1040. See also 3.377...×1038. 1.15868...×1042 = 64! / (32!×8!2×2!4×24) This is (a corrected value for) the number of possible chess positions, originally given by Shannon in the 1950 article "Programming a Computer for Playing Chess." (Phil. Mag. 41, 256-275). The formula is based on the idea that you can theoretically arrange all 32 pieces in any position whatsoever (giving 64!/32!) but that all pawns of a given colour are equivalent (8! for each colour), as is each pair of rooks (22) and each pair of knights (another 22); the bishops are not interchangeable but each has only 32 squares to choose from (24). However, this is inaccurate for a number of reasons. First and most important, a pawn cannot switch columns (ranks), or move past the opposing pawn in its rank, unless it captures. The more captures take place, the more flexibility the pawns have, but that decreases the number of pieces which decreases the number of board positions. Also, the possibility of pawn promotion increases the number of combinations somewhat. A far better estimate is that by John Tromp. The number of possible chess games is much higher. See also 765 and 2.081681...×10170.

20988936657440586486151264256610222593863921 = (2148+1)/17 ~= 2.098893665744×1043 In July 1951 Ferrier found this 44-digit prime using a mechanical desk calculator. It became the largest-known prime, breaking the record set by Lucas in 1876. This record did not stand long; it was broken by Miller and Wheeler in the same month. 34 63976656348486725806862358322168575784124416 ~= 6.397665...×1043 This is 447212, and is "nearly equal" to 398712 + 436512: it is a "near miss" for Fermat's Last Theorem. The numbers appear in the Simpsons episode "The Wizard of Evergreen Terrace". See also 8712. 1044 The value of the number called zài in Chinese. See also 104096. 393050634124102232869567034555427371542904832 ~= 3.9305×1044 This is 141×2141+1, the smallest number of the form n2n+1 that is prime. Cullen (the same one after whom the Cullen numbers are named) investigated numbers of this form in 1905.

824792557184288824246737061810550733633916929 = 3×(7×392-1)/2 ≈ 8.247925...×1044 This is a lower bound found by Milton Green for the value of BB(8), where BB(n) is the busy beaver function. 7.40119...×1045 = 7!×36 × 24! × 24!/246 = 7401196841564901869874093974498574336000000000 (The 4x4x4 Rubik's cube) The number of ways to arrange a 4×4×4 Rubik's Cube. The corner cubelets have the same number of combinations as the 2×2×2 cube (see 3674160). There are 24 edge pieces, which can be put in any of the 24!≈6.2×1023 permutations. There are 24 centre pieces — these would have 24! permutations, except for the fact that each of the four pieces of a given colour are indistinguishable from each other; so there are 24!/246 combinations for those pieces. See also 3674160, 4.3252...×1019, 2.8287...×1074, 1.5715...×10116, and 1.95005...×10160.

2(1.1×140) ≈ 2.28×1046 Randall Munroe made this estimate of the number of "meaningfully different" 140-character Twitter messages in English, using Shannon's estimate [137] of roughly 1.1 bits of information per letter (Twitter has since increased the Tweet length limit to 280, so it would now be about 5.21×1092). See also 2.45995..×10200, 10800, and 4×102254. 4.519364...×1046 An upper bound on the number of possible chess "diagrams" (a board configuration together with the facts of whose turn it is, who still has the option of castling, and any available en passant capture), computed by John Tromp. This estimate is better than that of Will Entriken and far better than mine. 5.4×1047 The amount of energy (in joules, or kg(m/s)^2) released in the "GW150914" event (merging of two co-orbiting black holes) detected by the LIGO gravitational wave experiment on 2015 Sep 14. This is 3 times the mass of the Sun times the speed of light squared. This amount of energy was released in less than 1 second, with a peak intensity about 50 times as great as all of the stars in the universe combined. 2054221614063184107682218077003539824552559296000 = 29×35×53×72×112×132×172×19×23×29×31×37×41×43×47×53×59×61×79×83×89×97×101 ≈ 2.054×1048 The smallest number that has at least 1010 distinct factors.

See also 12, 840, 45360, 720720, 3603600, 245044800, 278914005382139703576000 and 457936×10917. 2.393753...×1049 An upper bound on the number of possible chess "diagrams" (a board configuration together with the facts of whose turn it is, who still has the option of castling, and any available en passant capture), computed by Will Entriken in 2006. This estimate is more carefully considered than mine; the estimate by John Tromp is better still. 5.23198...×1049 (chess diagrams, by my estimate) This is a simple upper bound on the number of possible chess diagrams ("positions" together with the knowledge of whose turn it is, for whom castling is still permitted, and where en passant might occur). It is computed in a similar manner to Shannon's estimate of 1.15868...×1042. It allows between 2 and 32 pieces in play, with no more than 16 of one colour, including exactly one king of each colour, and up to 8 pawns of each colour (any of which might have been promoted to another piece). It is higher than Shannon's estimate because it allows pawn promotion, but is unrealistic because (among other reasons) one cannot promote all pawns without first capturing some other pieces. The number of possible chess games is much higher. See also 765, 8.065817...×1067 and 2.081681...×10170. 5.8535×1050 The length of a (Julian) year in Planck units. This is also the length of a light year in Planck length units. This can be used to convert the universe's age and size to Planck units. (Neither the age nor the size is known to sufficient precision for the discrepancy between the Julian year and other years, such as the tropical year, to make any difference.)

See also 5878625373183.6. 808017424794512875886459904961710757005754368000000000 = 246 × 320 × 59 × 76 × 112 × 133 × 17 × 19 × 23 × 29 × 31 × 41 × 47 × 59 × 71 ≈ 8.08...×1053 (the Monster group) This is the "order" (number of elements) in the largest sporadic finite simple group, called the "Monster group" or the Fischer-Griess group. (Some background: A "group" can be visualised as a set of transformations, e.g. rotations and reflections, that belong to an N-dimensional geometric structure such as a crystal lattice, or Rubik's Cube. A "simple" group has no "subgroups", which are subsets that themselves form a group; a "sporadic" group is one that does not fit into one of the infinite classes (cyclic, alternating, and Lie).) See also 196883. 1057 ± 20% There are approximately 1057 neutrons in a neutron star. Neutron stars (by current estimates) range anywhere from 1.35 to 2.1 times the mass of the sun. The mass of the Sun is about 1.99×1030 kg, and the number of neutrons in a kilogram of neutrons is about 1000×6.02×1023, so the number of neutrons in a neutron star ranges between about 8.1×1056 and 1.26×1057. The upper and lower mass limits are a bit uncertain, so we can safely just call it "1057 plus or minus 20 percent".

Since a neutron star is made up (almost) entirely of neutrons, and about half of these were protons that have combined with electrons via a nuclear interaction similar to electron capture, one could think of a neutron star as being the nucleus of element number N, where N is anywhere in this range near 1057. Using systematic element names, we could whimsically say that a "nucleus" of unniltripenthexhexoctuntriunquadunpentennbihexpenttrinilquad- hextritrisepttriseptoctniloctbiunseptununoctpentbihexquadhex- octbibihexnilseptseptennocthexenntrioctquadquadnilnilunium is a neutron star. 1059 Another large number that appears in the Lotus sutra texts of Mahayana Buddhism, where it appears as the word A-so-gi (あそぎ). See also 1011. 8.03×1060 (age of the Universe in Planck units) An approximate value for the age of the universe in Planck time units: r = 13.72×109 × 365.25 × 24 × 3600 / 5.391×10-44 ≈ 8.03×1060 For various reasons, this number is not equal to the "radius", nor is it exactly 1/3 the radius of the visible universe. However, for rough calculations of things like the current volume and space-time volume, and particularly for larger derived values like the number of alternate universes, it is more than adequate. See also 1040. 2.75×1061 (radius of the visible Universe in Planck units) An estimate of the radius of the "visible" universe in Planck length units. This is 46000000000 × 9.46×1015 / 1.616×10-35. It is not simply the "radius" in light-years corresponding to the age of the universe in years, for reasons explained in the universe size entry. It also accounts for changes in the rate of the universe's expansion, and the amount of its "curvature", according to the Lambda-CDM model.

See also 10122. 1063 (vigintillion) Archimedes, in his writing psammites (better known as The Sand Reckoner), estimated the size of the universe according to the heliocentric model of Artistarchus, and how many grains of sand would fit in it. He arrived at a value equivalent to one vigintillion, or 1063. Even more impressive, he described a system of numbers extending as high as 108×1016. Curiously, the number of protons in those 1063 grains of sand is very nearly equal to the number of protons in the visible universe (the Eddington Number), so Archimedes got the mass right even if he was a bit low regarding the volume. The word vigintillion is one of the number-names that had to be extrapolated by others based on the names established by Chuquet, and one of the few that appear in almost every dictionary; see this discussion, and see also 1033, 10303, 103003 and 103000003. Probably because of it being the largest number-word in the dictionary, H. P. Lovecraft used this number in two of his stories, including the 1926 Call of Cthulhu. (See also octillion). (Personal: For a while during 3rd grade this was the largest number I knew and on a few occasions I wrote it in the sand during recess: 1,000,000,000...,000,000. (counting out 21 sets of zeros). A mean kid would follow and wipe it out.) 2000000000000000000000000002000000000000000000000002000000002293 ≈ 2×1063 The "alphabetically last prime" found by Knuth and Miller's fall 1980 CS 204 class at Stanford[149].