Notable Properties of Specific Numbers


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1.0130653244...×10177 = 2588

The number of years in the longest time-period in the cosmology of Jainism, a religion and philosophy from India in the 6th century B.C. (From an article by J J O'Connor and E F Robertson)

8.45×10184

The current volume of the universe in Planck units. See also 8.02×1060, 1.69×10245, and 5.1843×1022652507173.

5.7324701932...×10207 = 75600000000000 × 840000028

Another number from measurement of time in Jainism. A purvi is 22×33×7×1011 = 7.56×1013 years, and a shirsha prahelika is 840000028 purvis, which works out to 5.7324701932...×10207 years.

1.267650600228...×10230 = 200100

Value of "googoc" from the Googology page. This page introduces a lot of novel number names and also summarizes a lot of other people's names for special large numbers.

The basic idea is to derive word endings, prefixes and infix parts by reverse etymology from existing names like "googol". According to that page, André Joyce noticed that googol is "googo" followed by the Roman numeral L (representing 50) and that googol=10100 is equal to 10050. He extrapolated this with the generalization that "googo" followed by any Roman numeral(s) z is (2z)z, which he chose to express as ack-h(2,ack-h(1,2,l),l) using the Herbert version of the Ackermann function. This means that "googoc" would be 200^100, "googod" is 1000500=101500, etc. In another similar generalization, anything followed by -ple- and Roman numeral(s) such as x is the result of raising x (or whatever) to the power of the thing to which -plex was added (perhaps he imagined "ple" stood for something like "placé dans l'exposant de"). Thus, for example, twoplex = two, 2, followed by -ple- and x representing 10, is 102; threepleiii is 33=27; and googodplem = 1000googod = 10001000500 = 103×101500.

1.69×10245

The four-dimensional volume (in space + time) of the known universe using the formula for the volume of a hypercone (with spherical cross-section) and the universe's age, expressed in Planck units. The hypercone has a 4-dimensional volume given by


V = (1/4 h) (4/3 π r3)

where h is the height of the hypercone and r is the radius of the sphere that forms the hypercone's base. h is the age of the universe in Planck time units and r is its current radius in Planck length units. Due to complexities of relativity and the way the universe expands, h and r are not the same. This gives 6.2×10243 for the 4-D volume.

Real models of the universe used by astrophysicists and cosmologists are much more complex and do not admit to such a simple calculation of volume, but most models would arrive at a figure close to this one.

See also 1.36×10403.

9.1197528826...×10262 = 20!×319 × 30!×229 × 60!/(5!12) × 60!/(5!12) × 60! / 25

This is the number of ways to arrange the pieces on a Gigaminx, a dodecahedron-shaped combinatorial puzzle with 230 movable pieces. The Gigaminx is to the Megaminx as the 5x5x5 cube is to the normal 3x3x3 Rubik's Cube. This video shows one in action, and here is another..

See also 1.0067×1068, 1.7989×10571, and 7.7263×10992.

10300

This is the base of the level-index representation used by Hypercalc. I chose it because it is close to the limit of the IEEE 754 double type. The bit of extra space above 10300 makes addition and roundoff algorithms much simpler.

It is also the basis of my ASCII-lexicographic ordering of arbitrarily high positive numbers, used internally by the source file of this web page. Here is an illustration of the system through examples:

 
ASCII representation

Value
0_0000000000000234 0.0000000000000234 = 2.34×10-14   (All 0's are shown, negative exponents in scientific notation are not supported. A big deficiency in the system, but it hasn't impacted me mainly because small numbers have never interested me much.)
0_1 0.1
9_34 9.34
a10_2 10.2   (Numbers with 2, 3, 4, or 5 digits use the letters a through d)
b256 256
c1 1000   (Trailing 0's can be left off.)
c1000 1000
c1729 1729
d19683 19683
e005_1 100000 = 1×105   (Numbers from 6 to 300 digits use e prefix, a 3-digit exponent and a mantissa)
e005_100 100000   (Extra 0's can be left off at the end provided it doesn't result in a different value from that which is intended.)
e005_1000007 100000.7
e005_13 130000   (Again, extra 0's left off)
e023_602 6.02×1023
e100_1 10100
e299_9 9×10299
p1_b300_0 1×10300   (Called "1 P.T. 300", which means "One Power of Ten, 300". The "300_0" part is the logarithm of the number being represented.)
p1_c2345_6 4×102345 ≅ 102345.6 = "1 P.T. 2345.6" (The ".6" is approzimately the logarithm to base 10 of 4. 10 to the power of 2345.6 is 4 times 10 to the power of 2345.)
p1_e009_345 103.45×109 = "1 P.T. 3.45×109"
p1_e299_999 109.99×10299 = "1 P.T. 9.99×10299"
p2_b300_0 1010300   (Called "2 P.T. 300" because it is "two powers of ten" with 300 above the two 10's)
pa10_c2345_6 10^(10^(...^(102345.6)))   (with a total of ten 10's. This is "10 P.T. 2345.6", that is, ten powers of 10 with 2345.6 at the top.)
pa99_c2345_6 99 P.T. 2345.6
pb100 100 P.T. 1   (100 powers of 10 with a 1 at the top. Since 101=10, the 1 can be ignored so it's just 100 powers of 10, and the "_1" can be left off.)
pb100_1 100 P.T. 1   (100 powers of 10 with a 1 at the top. Since 101=10, the 1 can be ignored so it's just 100 powers of 10.)
pb100_b234.5 100 P.T. 234.5
pb100_c1000 100 P.T. 1000
pb101_3 101 P.T. 3   (101 powers of 10 with a 3 at the top. Since 103=1000, this is the same as 100 P.T. 1000. This is the only potential ordering problem in the system. You have to be consistent, and not use "pb100_c1000" and "pb101_3" in the same collection of data.)
pb256_b619_299 256 P.T. 619.299   (255 powers of 10 with 1.99237...×10619 at the top. This is the large number called mega by Hugo Steinhaus and Leo Moser)
pd99999_c2345.6 99999 P.T. 2345.6
pe005_100 100000 P.T. 1   (Again, in this case, extra 0's can be left off at the end provided it doesn't result in a different value from that which is intended)
pe005_100000 100000 P.T. 1
pe005_100000_2 100000 P.T. 2   (However if the exponent at the top of the power-tower is anything other than 1 (in this case a "2"), then all the 0's in 100000 need to be shown.)
pe005_100000_c2345_6 100000 P.T. 2345.6
pe010_10000000000_2 10000000000 P.T. 2   =   (1010) P.T. 2 (Again, all 0's need to be shown in this case because there is a precise exponent at the top.)
pe299_9999999 (9.999999×10299) P.T. 1
pp1_b300_0 (10300) P.T. 1   (A power tower of 10's, of height 10300.)
pp1_e006_1 (101000000) P.T. 1   (A power tower of 10's, of height 101000000 = 10106.)
pp2_b300_0 (1010300) P.T. 1   (A power tower of 10's, of height 1010300.)
ppb27_1 (27 P.T. 1) P.T. 1   (A power tower of 10's, of height X, where X is a power tower of 10's of height 27).)
. . . (Contact me if you really need to go higher, but note you can't get much further before definitive ordering is nonobvious)

These ASCII strings can be used as labels, variables or function names in popular programming languages (including C, Perl and PHP) and are valid search keywords (for example, a Google search for "e023_602" finds this page).

10303

centillion, often said to be the largest number with a single-word name in English (and many other languages that use the Chuquet names).

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.

See also 3.4028236692093×1038, 1.1897314953572318×104932, 4.26448742×102525222 and 1.4403971939817846×10323228010.

1.397162914×10316

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 at on the MathWorld page85.

8.1847946207224960623437×10370

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.

1.36×10403

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 6.2×10340 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 Einstein's 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.

6.2×10340 is the single-perturbation count. Its factorial gives the total number of complete shufflings of the entire known universe's history.

See also 101016, 101.877×1054, 101077, 101082, 1010166, 103.67×10281, 1010375, 105.5×10405, 101010000000, 101010122, and 10101.51×103883775501690.

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

See also 10140 and 103.7218×1037.

10500

According to [104], 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.

See also 1040, 101016, 101.877×1054, 101077, 101082, 1010166, 103.67×10281, 1010375, 105.5×10405, 101010000000, 101010122, and 10101.51×103883775501690.

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

This is the number of ways to arrange the pieces on a Teraminx, a dodecahedron-shaped combinatorial puzzle with 530 movable pieces. Teraminx is to the Megaminx as the 7x7x7 cube is to the normal 3x3x3 Rubik's Cube. These puzzles have been made by dedicated hobbyists with access to CAD-CAM prototyping machines. Here is a video of one being assembled, and here you can see the puzzle being manipulated.

See also 1.0067×1068, 9.1197×10262, and 7.7263×10992.

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 457936006084633875 260691932542213506579481395376 080192442872707759996212114957 373537195900697943283211344130 969977204683723647091975242566 556807073476262370119366712949 612051508874565615465951982148 103948322515169952026557331614 199239782652240565877185274882 891122589783986489974588207230 026310073238799349251084594897 863556829085566422093207975001 895285824382289647389848615424 710629561529529589935914349946 023950287863307022313442880758 800532983282085207377266536998 146723331964258315488766981883 904240306133944424567760471103 539279962416731476757145320641 439420037963516042879919957607 890943287019373144639492683640 803862704805497501551907216898 677744138585826270309663329962 841518933729157858558919253022 063551926057138672786596389094 200184031909805595086778342937 081605771699885426749776777391 919555685119629369584896777148 250878775274042686107865894781 763500774758450843791837394393 056896301600021929961984000000. The factorization of this number, along with the other record setters up to 103535, was found by Achim Flammenkamp7. See also 12, 840, 45360, 720720, 3603600, 245044800, 278914005382139703576000 and 2054221614063184107682218077003539824552559296000.

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

This is the number of ways to arrange the pieces on a Petaminx, a dodecahedron-shaped combinatorial puzzle with 950 movable pieces. Petaminx is to the Megaminx as a 9x9x9 cube would be to the normal 3x3x3 Rubik's Cube, if such a thing existed. Believe it or not, someone actually built this puzzle using parts cast from an industrial prototyping machine, and sold it online for over $3000 U.S. A video of the puzzle being used can be seen here.

See also 1.0067×1068, 9.1197×10262, and 1.7989×10571.

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.

1.53×101165

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.

1.90797007527×101280 = 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?"

2.85542542232×101331 = 24423-1

M4423, the 20th Mersenne Prime; see 1.90797007527×101280.

4.6×101439

As of early 2008, the record 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.

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.

2.5×102879

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 for more.


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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    5×1011    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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