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

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 *n*th 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)

Return to table of contents.

**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 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

- (#??)
**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

Return to table of contents.

(These files are updated approximately once a month.) The following related files are also available:

- 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).

Finger primes@math.utm.edu for other ways to receive these files.

- types.txt
- Explains some of the different types of primes on the list (palindrome, Mersenne...).
- mersenne.txt
- A short note about the known Mersennes.

Return to table of contents.

- 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)

Return to table of contents.

**a^b**,**a+/-b**- Here
**a^b**is a raised to the b*th*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.

Suppose that there exist only finitely many primesQuoted from page 4 of Ribenboim's "Book of Prime Number Records" mentioned above. Ribenboim's book contains "nine and a half" proofs of this theorem!p1<p2< ... <pr. LetN= (p1)(p2)...(pr) > 2. The integerN-1, being a product of primes, has a prime divisorpiin common withN; so,pidividesN- (N-1) =1, which is absurd!

Return to table of contents.

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!)

Return to table of contents.

[ Chris Caldwell | The Back Door ]