## Mersenne Prime

A Mersenne Number which is Prime is called a Mersenne prime. In order for the Mersenne number defined by

for an Integer to be Prime, must be Prime. This is true since for Composite with factors and , . Therefore, can be written as , which is a Binomial Number and can be factored. Every Mersenne Prime gives rise to a Perfect Number.

If is a Prime, then Divides Iff is Prime. It is also true that Prime divisors of must have the form where is a Positive Integer and simultaneously of either the form or (Uspensky and Heaslet). A Prime factor of a Mersenne number is a Wieferich Prime Iff , Therefore, Mersenne Primes are not Wieferich Primes. All known Mersenne numbers with Prime are Squarefree. However, Guy (1994) believes that there are which are not Squarefree.

Trial Division is often used to establish the Compositeness of a potential Mersenne prime. This test immediately shows to be Composite for , 23, 83, 131, 179, 191, 239, and 251 (with small factors 23, 47, 167, 263, 359, 383, 479, and 503, respectively). A much more powerful primality test for is the Lucas-Lehmer Test.

It has been conjectured that there exist an infinite number of Mersenne primes, although finding them is computationally very challenging. The table below gives the index of known Mersenne primes (Sloane's A000043) , together with the number of digits, discovery years, and discoverer. A similar table has been compiled by C. Caldwell. Note that the region after the 35th known Mersenne prime has not been completely searched, so identification of the'' 36th Mersenne prime is tentative. L. Welsh maintains an extensive bibliography and history of Mersenne numbers. G. Woltman has organized a distributed search program via the Internet in which hundreds of volunteers use their personal computers to perform pieces of the search.

 # Digits Year Published Reference 1 2 1 Antiquity 2 3 1 Antiquity 3 5 2 Antiquity 4 7 3 Antiquity 5 13 4 1461 Reguis 1536, Cataldi 1603 6 17 6 1588 Cataldi 1603 7 19 6 1588 Cataldi 1603 8 31 10 1750 Euler 1772 9 61 19 1883 Pervouchine 1883, Seelhoff 1886 10 89 27 1911 Powers 1911 11 107 33 1913 Powers 1914 12 127 39 1876 Lucas 1876 13 521 157 1952 Lehmer 1952-3, Robinson 1952 14 607 183 1952 Lehmer 1952-3, Robinson 1952 15 1279 386 1952 Lehmer 1952-3, Robinson 1952 16 2203 664 1952 Lehmer 1952-3, Robinson 1952 17 2281 687 1952 Lehmer 1952-3, Robinson 1952 18 3217 969 1957 Riesel 1957 19 4253 1281 1961 Hurwitz 1961 20 4423 1332 1961 Hurwitz 1961 21 9689 2917 1963 Gillies 1964 22 9941 2993 1963 Gillies 1964 23 11213 3376 1963 Gillies 1964 24 19937 6002 1971 Tuckerman 1971 25 21701 6533 1978 Noll and Nickel 1980 26 23209 6987 1979 Noll 1980 27 44497 13395 1979 Nelson and Slowinski 1979 28 86243 25962 1982 Slowinski 1982 29 110503 33265 1988 Colquitt and Welsh 1991 30 132049 39751 1983 Slowinski 1988 31 216091 65050 1985 Slowinski 1989 32 756839 227832 1992 Gage and Slowinski 1992 33 859433 258716 1994 Gage and Slowinski 1994 34 1257787 378632 1996 Slowinski and Gage 35 1398269 420921 1996 Armengaud, Woltman, et al. 36? 2976221 895832 1997 Spence 37? 3021377 909526 1998 Clarkson, Woltman, et al.

See also Cunningham Number, Fermat-Lucas Number, Fermat Number, Fermat Number (Lucas), Fermat Polynomial, Lucas-Lehmer Test, Mersenne Number, Perfect Number, Repunit, Superperfect Number

References

Bateman, P. T.; Selfridge, J. L.; and Wagstaff, S. S. The New Mersenne Conjecture.'' Amer. Math. Monthly 96, 125-128, 1989.

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 66, 1987.

Beiler, A. H. Ch. 3 in Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. New York: Dover, 1966.

Caldwell, C. Mersenne Primes: History, Theorems and Lists.'' http://www.utm.edu/research/primes/mersenne.shtml.

Caldwell, C. GIMPS Finds a Prime! is Prime.'' http://www.utm.edu/research/primes/notes/1398269/.

Colquitt, W. N. and Welsh, L. Jr. A New Mersenne Prime.'' Math. Comput. 56, 867-870, 1991.

Conway, J. H. and Guy, R. K. Mersenne's Numbers.'' In The Book of Numbers. New York: Springer-Verlag, pp. 135-137, 1996.

Gillies, D. B. Three New Mersenne Primes and a Statistical Theory.'' Math Comput. 18, 93-97, 1964.

Guy, R. K. Mersenne Primes. Repunits. Fermat Numbers. Primes of Shape [sic].'' §A3 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 8-13, 1994.

Haghighi, M. Computation of Mersenne Primes Using a Cray X-MP.'' Intl. J. Comput. Math. 41, 251-259, 1992.

Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 14-16, 1979.

Kraitchik, M. Mersenne Numbers and Perfect Numbers.'' §3.5 in Mathematical Recreations. New York: W. W. Norton, pp. 70-73, 1942.

Kravitz, S. and Berg, M. Lucas' Test for Mersenne Numbers .'' Math. Comput. 18, 148-149, 1964.

Lehmer, D. H. On Lucas's Test for the Primality of Mersenne's Numbers.'' J. London Math. Soc. 10, 162-165, 1935.

Leyland, P. ftp://sable.ox.ac.uk/pub/math/factors/mersenne.

Mersenne, M. Cogitata Physico-Mathematica. 1644.

Mersenne Organization. GIMPS Discovers 36th Known Mersenne Prime, is Now the Largest Known Prime.'' http://www.mersenne.org/2976221.htm.

Mersenne Organization. GIMPS Discovers 37th Known Mersenne Prime, is Now the Largest Known Prime.'' http://www.mersenne.org/3021377.htm.

Noll, C. and Nickel, L. The 25th and 26th Mersenne Primes.'' Math. Comput. 35, 1387-1390, 1980.

Powers, R. E. The Tenth Perfect Number.'' Amer. Math. Monthly 18, 195-196, 1911.

Powers, R. E. Note on a Mersenne Number.'' Bull. Amer. Math. Soc. 40, 883, 1934.

Sloane, N. J. A. Sequence A000043/M0672 in An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

Slowinski, D. Searching for the 27th Mersenne Prime.'' J. Recreat. Math. 11, 258-261, 1978-1979.

Slowinski, D. Sci. News 139, 191, 9/16/1989.

Tuckerman, B. The 24th Mersenne Prime.'' Proc. Nat. Acad. Sci. USA 68, 2319-2320, 1971.

Uhler, H. S. A Brief History of the Investigations on Mersenne Numbers and the Latest Immense Primes.'' Scripta Math. 18, 122-131, 1952.

Uspensky, J. V. and Heaslet, M. A. Elementary Number Theory. New York: McGraw-Hill, 1939.

Weisstein, E. W. Mersenne Numbers.'' Mathematica notebook Mersenne.m.

Welsh, L. Marin Mersenne.'' http://www.scruznet.com/~luke/mersenne.htm.

Welsh, L. Mersenne Numbers & Mersenne Primes Bibliography.'' http://www.scruznet.com/~luke/biblio.htm.

Woltman, G. The GREAT Internet Mersenne Prime Search.'' http://www.mersenne.org/prime.htm.