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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.

1.90797007527×10¹²⁸⁰ = 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.

10⁴⁰⁹⁶

The value of the number called zài in one ancient Chinese system for naming large numbers³⁶. In this system, The successive names yì, zhào, etc. name successive squares of wàn (which is 10⁴), thus yì=10⁸, zhào=10¹⁶, and so on up to zài=10⁴⁰⁹⁶. In modern usage, zài is "merely" 10⁴⁴. See also 10000.

1.1897314953572318×10⁴⁹³² ≅ 2¹⁶³⁸⁴ = 2²¹⁴

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.

See also 3.4028236692093×10³⁸, 1.797693134862×10³⁰⁸, 1.9488283827×10²⁹⁶⁰³, 4.26448742×10^2525222 and 1.4403971939817846×10^323228010.

10⁵⁰⁰⁰

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.

4.486791...×10⁶⁵³² = 2²¹⁷⁰¹ - 1

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 M_n for primeness by computing the sequence S₁ = 4, S_n+1 = S_n² - 2, and checking if S_n-1 divides evenly into M_n. If it does, M_n is prime.

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 was 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.

10¹⁵⁰⁰⁰

As of 2007, this is the approximate limit on the size of numbers that can be shown to be composite using probabilistic primality tests. Such tests show that a number is either composite, or very probably prime (i.e. with probability a tiny bit less than 1.000).

See also 10⁵⁰⁰⁰.

5.19344195...×10¹⁵⁰⁷⁰ = 2638⁴⁴⁰⁵ + 4405²⁶³⁸

This 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 now called Leyland numbers in his honor. 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 Broadhurst 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.

2.0035299...×10¹⁹⁷²⁸ = 2⁶⁵⁵³⁶-3

The Ackermann function is a function that grows very fast, but has a surprisingly innocent-looking definition. Using the two-argument version of Peters, A(m,n) = n+1 (if m=0) or A(m-1,1) (if m>0 and n=0):> or A(m-1,A(m,n-1)) <$:(for remaining cases). This function produces the following table:

The first row is the positive integers, and each subsequent row is an Nth-term sequence generated from the row before it. Row 2 is linear, row 3 is exponential, and row 4 grows like the higher hyper4 operator.

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 2^B +/- 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.

1.9488283827...×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.

See also 3.4028236692093×10³⁸, 1.797693134862×10³⁰⁸, 1.1897314953572318×10⁴⁹³², 4.26448742×10^2525222 and 1.4403971939817846×10^323228010.

10⁴⁰⁰⁰⁰ = 10000¹⁰⁰⁰⁰

The value of a myriad to the power of itself, written (by the system of Apollonius of Perga) as a little M directly above a larger M. Significantly larger values 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.

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

As of early 2007, 2003663613×2¹⁹⁵⁰⁰⁰-1 and 2003663613×2¹⁹⁵⁰⁰⁰+1 were the largest known pair of twin primes.

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 this record twin prime, 4(n-1)! is about 10^{8.3116888×1058714}.

5.05640407...×10⁷⁸³²⁷ = 2²⁶⁰¹⁹⁹

See 2.74858523...×10⁸⁰⁵⁸⁸.

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, it was changed to 2²⁶⁰¹⁹⁹.) It is one of the largest numbers used in a work of fiction. (See also 42.) 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.

7.760271406486818269530232833213...×10²⁰²⁵⁴⁴

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 dfferent colors — 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 color, you would not be called unskilled or ignorant of numbers, but not yet shalt thou be numbered among the wise. But understand also these conditions: [The white bulls could stand together with the black bulls in rows, such that the number of cattle in each row was equal and that number was equal to the total number of rows, thus forming a perfect square. And the yellow bulls could stand together with the dappled bulls, with a single bull in the first row, two in the second row, and continuing steadily to complete a perfect triangle.] If thou art able, O stranger, to find out all these things and gather them together in your mind, giving all the relations, thou shalt depart crowned with glory and knowing that thou hast been adjudged perfect in this species of wisdom.

If you solve just the first part of the problem, the smallest solution for the total number of cattle is 50389082. But if you add the additional two constraints in the second part, the solution is much higher — about 7.76×10²⁰²⁵⁴⁴. It took until 1880 to find this answer, published by Amthor.

In 1931, in a letter to the New York Times, it was written

Since it has been calculated that it would take the work of a thousand men for a thousand years to determine the complete [exact] number [of cattle], it is obvious that the world will never have a complete solution.

of course, digital computers made the exact calculation possible, and the number was first calculated in 1965 by Williams, German and Zarnke on an IBM 7040. The 202545-digit number was first published in 1981 by Nelson. In 1998, Vardi showed that the number was the value of

25194541/184119152 × (109931986732829734979866232821433543901088049 + 50549485234315033074477819735540408986340 √4729494) ⁴⁶⁵⁸

rounded up to the nearest integer. In 2001, Nygrén showed how the problem could be solved in a manner simple enough (perhaps) to be known to the ancients (although it would not have enabled them to actually calculate the value of the solution, just prove that there is a solution and show how to calculate it).

(Reference: Chris Rorres' pages on Archimedes)

See also See also 3121.

8.147175681×10⁴²⁰⁹²⁰ = 2^1398269-1

This is the first Mersenne prime found by a participant in the GIMPS (Great Internet Mersenne Prime Search) project. It was discovered by Armengaud in 1996.

2.06506356×10^1262611 = 2²²²-1

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

1.95603995×10^1834097 = 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 10⁴⁰⁰⁰⁰, 2.748×10⁸⁰⁵⁸⁸ and 3.196×10²⁸²³⁰³.

4.370757...×10^2098959 = 2^6972593 - 1

Record-holder for largest known prime, before being superceded by 2^13466917-1.

4.26448742×10^2525222 = 2²²³

This is (approximately) the maximum value that can be represented in the floating-point format used by PARI, the free open-source symbolic math package developed at University Bordeaux, France.

See also 3.4028236692093×10³⁸, 1.797693134862×10³⁰⁸, 1.1897314953572318×10⁴⁹³² and 1.4403971939817846×10^323228010.

3.196×10²⁸²³⁰³ = 35¹⁸²⁸³¹

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 1.95×10^1834097. (Note: this value used to be listed under 6.8738×10⁴¹⁶⁸⁹ = 35²⁷⁰⁰⁰ and atributed 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.)

10^3000003 = 10^3×106+3

This number has the somewhat contrived name "milli-millillion". It is the largest example I have seen of a number name in the Latin-prefix system which includes the common names billion, trillion, vigintillion, etc.

9.2494778×10^4053945 = 2^13466917 - 1

From late 2001 until 2003 Nov 17 this was the largest known prime number and the largest known Mersenne prime. It was discovered by Michael Cameron, a member of the GIMPS (Great Internet Mersenne Prime Search) project. See this list of all known Mersenne primes.

1.259769×10^6320429 = 2^20996011 - 1

Discovered on 2003 Nov 17, and until 2004 May 15 was the largest known prime number and the largest known Mersenne prime. It was discovered by Michael Shafer, a member of the GIMPS (Great Internet Mersenne Prime Search) project. See this list of all known Mersenne primes.

2.99410453×10^7235732 = 2^24036583 - 1

Discovered on 2004 May 15, and and until 2005 Feb 18 was the largest known prime number and the largest known Mersenne prime. It was discovered by Josh Findley, a member of the GIMPS (Great Internet Mersenne Prime Search) project. See this list of all known Mersenne primes.

1.2216464×10^7816229 = 2^25964951 - 1

Discovered on 2005 Feb 18, and until 2005 Dec 15 was the largest known prime number and the largest known Mersenne prime. It was discovered by Dr. Martin Nowak, a member of the GIMPS (Great Internet Mersenne Prime Search) project. See this list of all known Mersenne primes.

4.27764198×10^8107891 = 2^13466916(2^13466917-1)

For a while this was the largest known perfect number. See here for a complete list.

3.15416507×10^9152051 = 2^30402457-1

Discovered on 2005 Dec 15, and until 2006 Sep 4 was the largest known prime number and the largest known Mersenne prime. It was discovered by Dr. Curtis Cooper and Dr. Steven Boone, members of the GIMPS (Great Internet Mersenne Prime Search) project. See this list of all known Mersenne primes.

1.24575039×10^9808357 = 2^32582657 - 1

Discovered on 2006 September 4, and currently the largest known prime number and the largest known Mersenne prime. It was discovered by Dr. Curtis Cooper and Dr. Steven Boone, members of the GIMPS (Great Internet Mersenne Prime Search) project. See this list of all known Mersenne primes.

4.4823309×10^14471464 = 2^24036582(2^24036583-1)

As of 2004 May 15, the largest known perfect number. See here for a complete list.

1.4403971939817846×10^323228010 ≅ 2^1073740208 = 2^(230-1616)

This is (approximately) the maximum value that can be represented in the floating-point format used by Mathematica^TM, the symbolic mathametics program by Wolfram Research. The format uses a 31-bit exponent field. I know of no standard (IEEE or otherwise) floating-point format that uses a 31-bit exponent. This is also the largest exponent field of any exponent format I have found (however, Hypercalc achieves a far greater range than any conceivable floating-point format by representing numbers in a different way).

See also 3.4028236692093×10³⁸, 1.797693134862×10³⁰⁸, 1.1897314953572318×10⁴⁹³² and 4.26448742×10^2525222.

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