The Largest Known Primes

1. Introduction (What are primes? Who cares?)
2. The Top Ten Record Primes:
       largest, twin, Mersenne, primorial&factorial, and Sophie Germain
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?

  1. 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 (see the document How Many?)

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 Lagrange's Theorem from group theory. See the separate documents on 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 20,000 primes is available in several forms below; but first we offer a few records for your perusal. (See: notation below 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)

  2. The "Top Ten" Record Primes

The Ten 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 largest primes and their discoverers are as follows.

• 2^859433-1 (258716 digits); Slowinski & Gage, 1994
• 2^756839-1 (227832 digits); Slowinski & Gage, 1992 [Peterson92]
• 391581*2^216193-1 (65087 digits); Amdahl Six, 1989 [BNPSSZ90]
• 2^216091-1 (65050 digits); Slowinski, 1985
• 3*2^157169+1 (47314 digits); Jeffrey Young, 1995 [Young**]
• 9*2^149143+1 (44898 digits); Jeffrey Young, 1995 [Young**]
• 9*2^147073+1 (44275 digits); Jeffrey Young, 1995 [Young**]
• 9*2^145247+1 (43725 digits); Jeffrey Young, 1995 [Young**]
• 2^132049-1 (39751 digits); Slowinski, 1983
• 9*2^127003+1 (38233 digits); Jeffrey Young, 1995 [Young**]

Click here to see the one hundred largest known primes.

The Ten 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). Recently there has been keen competition for the record, but Indlekofer and Ja'rai new record may stand for awhile.

• 242206083*2^38880+/-1 (11713 digits) Indlekofer and Ja'rai, Nov. 1995
• 570918348*10^5120+/-1 (5129 digits); Dubner, Oct. 1995 [Ribenboim95 p263]
• 697053813*2^16352+/-1 (4932 digits); Indlekofer and Ja'rai 1994 [IJ96]
• 6797727*2^15328+1/-1 (4622 digits); Tony Forbes 1995
• 1692923232*10^4020+/-1 (4030 digits); Dubner 1993 [IJ96, Peterson93]
• 4655478828*10^3429+/-1 (3439 digits); Dubner 1993 [IJ96]
• 1706595*2^11235+/-1 (3389 digits); The Amdahl Six, 1989 [PSZ90]
• 459*2^8529+/-1 (2571 digits); Dubner 1993
• 1171452282*10^2490+1 (2500 digits); Dubner 1991
• 571305*2^7701+1 (2324 digits); The Amdahl Six, 1989 [PSZ90]

Click here to see all of the known twin primes with at least 1000 digits.

The Ten 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 [Peterson92]
• (#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 [CW91]
• (#28) 2^86243-1 (25962 digits); Slowinski, 1982 [Ewing83]
• (#27) 2^44497-1 (13395 digits); Nelson and Slowinski, 1979 [Slowinski79]
• (#26) 2^23209-1 (6987 digits); Noll, 1979 [NN80]
• (#25) 2^21701-1 (6533 digits); Noll & Nickel, 1978 [NN80]
• (#24) 2^19937-1 (6002 digits); Tuckerman, 1971 [Tuckerman71]

See our page on Mersenne numbers for more information including a complete table of the known Mersennes.

The Ten Largest Known Factorial and Primorial Primes

Euclid's proof that there are infinitely many primes uses numbers of the form n#+1. Kummer's proof uses those of the form n#-1. Sometime students look at these proofs and assume the numbers n#+/-1 are always prime, but that is not so. 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 [Caldwell95]
• 3507!-1 (10912 digits); Caldwell 1992 [Caldwell95]
• 24029#+1 (10387 digits); Caldwell 1993 [Caldwell95]
• 23801#+1 (10273 digits); Caldwell 1993 [Caldwell95]
• 18523#+1 (8002 digits); Dubner 1989 [Dubner89a]
• 15877#-1 (6845 digits); Caldwell and Dubner 1992 [Caldwell95]
• 13649#+1 (5862 digits); Dubner 1987 [Dubner87]
• 1963!-1 (5614 digits); Caldwell and Dubner 1992 [Caldwell95]
• 13033#-1 (5610 digits); Caldwell and Dubner 1992 [Caldwell95]
• 11549#+1 (4951 digits); Dubner 1986 [Dubner87]
• 1477!+1 (4042 digits); Dubner 1984 [DD85]

  3. The Complete List of the Largest Known Primes

As a searchable database

You may search the list by keyword, number size, discoverer...

all.txt

The whole list! This is a large file: 1M.

all.zip

The whole list (all.txt) pkZipped, so it is roughly one fourth the size of all.txt: 279K.

short.txt

All of the gigantic primes (those with 10,000 digits or more), plus the 'interesting' smaller primes (that is, those with comments on the list). So this is a much smaller file: 71K.

  4. Other Sources of Large Primes

• Paulo Ribenboim, "The New Book of Prime Number Records," Springer-Verlag, New York, 1995 (QA246 .R472) [Ribenboim95].
• Paulo Ribenboim, "The Little Book of Big Primes," Springer-Verlag, 1991 (QA246 .R47) [Ribenboim91]. (A less mathematical version of the above text.)
• Hans Riesel, "Prime Numbers and Computer Methods for Factorization," Progress in Mathematics, BirkhΣuser Boston, vol. 126, 1994 [Riesel94].

• 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 [BLSTW88].

See The Prime Page for many other web sources.

  5. 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.

p# ("p primorial")

If p is any prime, then p# is the product of all the primes less than or equal to p, so 5#=2*3*5=30.

Titanic Primes

Samuel Yates defined a titanic prime to be any prime with at least 1,000 digits. When he indroduced this term [Yates84] there were only 110 such primes known; now there are over 150 times that many!

Gigantic Primes

A gigantic prime is a prime with at least 10,000 digits. This term was also introduced by Samuel Yates [Yates92b].

  6. Euclid's Proof of the Infinitude of Primes

Suppose that there exist only finitely many primes p₁ < p₂ < ... < p_r.   Let N = p₁p₂...p_r > 2.   The integer N-1, being a product of primes, has a prime divisor p_i in common with N; so, p_i divides N - (N-1) =1, which is absurd!

  7. Comments, Suggestions? New records?