The Largest Known Primes

Contents:

1. Introduction (What are primes? Who cares?)
2. The "Top Ten" Record Primes
3. The Complete List of the Largest Known Primes
4. Other Sources of Prime Information
5. Notation and Definitions
6. Euclid's Proof of the Infinitude of Primes
7. Comments? Suggestions? New records? New Links?

Related Local Documents:

• Finding (and Proving) Primes.

Introduction

The ancient Greeks proved (ca 300 BC) that there were infinitely many primes and that they were irregularly spaced (there can be arbitrarily large gaps between successive primes). On the other hand, in the nineteenth century it was shown that the number of primes less than or equal to n approaches n/(ln n) (as n gets very large); so a rough estimate for the nth prime is n ln n.

The Sieve of Eratosthenes is still the most efficient way of finding all very small primes (e.g., those less than 1,000,000). However, most of the largest primes are found using special cases of Largrange's Theorem from group theory. See the separate document finding primes for more information.

Recently computers and cryptology have given a new emphasis to search for ever larger primes--at this site we store lists of thousands of these record breaking primes, all of which have over 1,000 digits! The complete list of approximately 10,000 primes is available in several forms below; but first we offer a few records for your perusal. (See: notation for an explanation of the symbols used here; references for a list of other sources of prime information; and of course, comments and suggestions are always requested!)

The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length... Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated. (Karl Friedrich Gauss, Disquisitiones Arithmeticae, 1801)

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The "Top Ten" Record Primes

The Five Largest Known Primes

On January 4, 1994 David Slowinski announced on the internet that he and Paul Gage have found a new record prime: 2^859433-1. The proof of this 258,716 digit number's primality (using the traditional Lucas-Lehmer test) took about 7.2 hours on a Cray C90 super computer. Richard Crandall independently verified the primality. (The complete decimal expansion of this prime is available here). The next largest primes and their discoverers are

• 2^756839-1 (227832 digits); Slowinski and Gage, 1992
• 391581*2^216193-1 (65087 digits); Noll and others, 1989
• 2^216091-1 (65050 digits); Slowinski, 1985
• 3*2^157169+1 (47314 digits); Jeffrey Young, 1995 (Fermat F157167 Factor)
• 9*2^149143+1 (44898 digits); Jeffrey Young, 1995
• 9*2^147073+1 (44275 digits); Jeffrey Young, 1995
• 9*2^145247+1 (43725 digits); Jeffrey Young, 1995
• 2^132049-1 (39751 digits); Slowinski, 1983
• 9*2^127003+1 (38233 digits); Jeffrey Young, 1995

(The primes by Jeffrey Young are from a recent preprint.)

The Largest Known Twin Primes

Twin primes are primes of the form p and p+2, i.e., they differ by two. It is conjectured, but not yet proven, that there are infinitely many twin primes (the same is true for all of the following forms of primes, see conjectures for more information).

• 697053813*2^16352+/-1 (4932 digits); Indlekofer and Ja'rai 1994
• 6797727*2^15328+1/-1 (4622 digits); Tony Forbes
• 1692923232*10^4020+/-1 (4030 digits); Dubner 1993
• 4655478828*10^3429+/-1 (3439 digits); Dubner 1993
• 1706595*2^11235+/-1 (3389 digits); Noll and others 1989
• 459*2^8529+/-1 (2571 digits); Dubner 1993

The Largest Known Mersenne Primes

Mersenne primes are primes of the form 2^p-1. These are the easiest type of number to check for primality on a binary computer so they usually are also the largest primes known. Altogether 33 of these primes are known, but since the region between the largest two and the previous primes has not been completely searched we do not know if the largest two are the 32nd and 33rd according to size.

• (#??) 2^859433-1 (258716 digits); Slowinski and Gage, 1994
• (#??) 2^756839-1 (227832 digits); Slowinski and Gage, 1992
• (#31) 2^216091-1 (65050 digits); Slowinski, 1985
• (#30) 2^132049-1 (39751 digits); Slowinski, 1983
• (#29) 2^110503-1 (33265 digits); Colquitt and Welsh, 1988
• (#28) 2^86243-1 (25962 digits); Slowinski, 1983

See mersenne.txt for more information on Mersenne primes.

The Largest Known Factorial and Primorial Primes

Eulcid's proof that there are infinitely many primes used numbers of the form n#+1. When numbers of the form n#+/-1 are prime they are called primorial primes. Similarly numbers of the form n!+/-1 are called factorial primes. The current record holders and their discoverers are:

• 3610!-1 (11277 digits); Caldwell 1993
• 1477!+1 (4042 digits); Dubner 1984
• 24029#+1 (10387 digits); Caldwell 1993
• 15877#-1 (6845 digits); Caldwell and Dubner 1992

See types.txt for descriptions of these and other special forms of primes.

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The Complete List of the Largest Known Primes

all.txt

The whole list. This is a very long list of about 10,000 primes in a text file about 500k bytes long.

all.zip

The whole list (all.txt) pkZipped, so it is roughly one fourth the size of all.txt.

short.txt

All of the largest primes (those with 5,000 digits or more), plus the 'interesting' smaller primes (that is, those with comments on the list).

types.txt

Explains some of the different types of primes on the list (palindrome, Mersenne...).

mersenne.txt

A short note about the known Mersennes.

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Other Sources of Large Primes

• Paulo Ribenboim, "The Book of Prime Number Records," Springer-Verlag, 1988 (QA246 .R472).
• Paulo Ribenboim, "The Little Book of Big Primes," Springer-Verlag, 1991 (QA246 .R47).
• Hans Riesel, "Prime Numbers and Computer Methods for Factorization," Progress in Mathematics, Birkhäuser Boston, vol. 57, 1985; and vol 126, 1994.
• D. M. Bressoud, "Factorization and Primality Testing," Undergraduate Texts in Mathematics, Springer-Verlag, 1989.
• Henri Cohen, "A Course in Computational Algebraic Number Theory" Springer-Verlag, Graduate texts in mathematics, vol. 138 (1993).

• J. Brillhart, et al., "Factorizations of b^n±1, b = 2,3,5,6,7,10,11,12 up to high powers," American Mathematical Society, 1988.

Other Web Sources:

• The section on primes from the excellent Frequently Asked Questions in Mathematics (from the sci.math usenet group, Alex Lopez-Ortiz editor).
• Yahoo's Mathematics Resource Listing
• A General Historical Introduction to Primes (Linked to Excellent Biographies)

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Notation and Definitions

a^b, a+/-b

Here a^b is a raised to the bth power, that is a times itself b times (so 2^4 = 16); and a+/-b is a "plus or minus b".

n! ("n factorial")

If n is an integer, then n! is the product of all the positive integers less than or equal to n, so 5!=1*2*3*4*5=120.

n# ("n primorial")

If n is an integer, then n# is the product of all the primes less than or equal to n, so 6#=5#=2*3*5=30.

Titanic Primes

A titanic prime is a prime with at least 1,000 digits. Samuel Yates introduced this term in 1984 ("Sinkers of the Titanic", J. Recreational Math. 17, 1984/5, p268-274) when there were only 110 such primes known; now there are about 10,000 known!

Gigantic Primes

A gigantic prime is a prime with at least 10,000 digits. This term was also introduced by Samuel Yates.

Euclid's Proof of the Infinitude of Primes

Suppose that there exist only finitely many primes p1 < p2 < ... < pr. Let N = (p1)(p2)...(pr) > 2. The integer N-1, being a product of primes, has a prime divisor pi in common with N; so, pi divides N - (N-1) =1, which is absurd!

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Comments, Suggestions? New records?

Chris Caldwell (e-mail to: caldwell@utm.edu or primes@utm.edu)

(Do you know about other sources of prime information? If so, then please let me know!)

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