# Mersenne Primes

N is an even perfect number if and only if

### ```N = 2^(q-1) * (2^q - 1) and 2^q - 1 is prime ```

It should also be noted that for 2^q - 1 to be prime q must be prime. So when we search for even perfect numbers, we search on q equal to the primes.

The numbers M(q) = 2^q - 1 (with q prime) are called Mersenne numbers. If M(q) = is prime then it is called a Mersenne prime. If a prime q makes a Mersenne number a Mersenne prime, then P(q) = 2^(q-1) * (2^q - 1) is a Perfect number.

Here are the 47 known Mersenne primes, M(q), as of April 12, 2009, listed by size:

### ``` nr q digits digits Year Discoverer (Computer) in M(q) in P(q) yyyy-mm-dd -- ------- ------- ------- ---------- ---------------------- 1 2 1 1 500BC? - 2 3 1 2 500BC? - 3 5 2 3 275BC? - 4 7 3 4 275BC? - 5 13 4 8 1456 Anonymous 6 17 6 10 1588 Pietro Antonio Cataldi 7 19 6 12 1588 Pietro Antonio Cataldi 8 31 10 19 1772 Leonhard Euler 9 61 19 37 1883 I. M. Pervushin 10 89 27 54 1911 R. E. Powers 11 107 33 65 1914 R. E. Powers 12 127 39 77 1876 Francois Edouard Anatole Lucas 13 521 157 314 1952-01-30 Raphael M. Robinson (SWAC) 14 607 183 366 1952-01-30 Raphael M. Robinson (SWAC) 15 1279 386 770 1952-06-26 Raphael M. Robinson (SWAC) 16 2203 664 1327 1952-10-07 Raphael M. Robinson (SWAC) 17 2281 687 1373 1952-10-09 Raphael M. Robinson (SWAC) 18 3217 969 1937 1957-09-08 Hans Riesel (BESK) 19 4253 1281 2561 1961-11-03 Alexander Hurwitz & John L. Selfridge (IBM 7090) 20 4423 1332 2663 1961-11-03 Alexander Hurwitz & John L. Selfridge (IBM 7090) 21 9689 2917 5834 1963-05-11 Donald B. Gillies (ILLIAC 2) 22 9941 2993 5985 1963-05-16 Donald B. Gillies (ILLIAC 2) 23 11213 3376 6751 1963-06-02 Donald B. Gillies (ILLIAC 2) 24 19937 6002 12003 1971-03-04 Bryant Tuckerman (IBM 360/91) 25 21701 6533 13066 1978-10-30 Landon Curt Noll & Laura A. Nickel (Cyber 174) 26 23209 6987 13973 1979-02-09 Landon Curt Noll (Cyber 174) 27 44497 13395 26790 1979-04-08 David Slowinski & Harry L. Nelson (Cray 1) 28 86243 25962 51924 1982-09-25 David Slowinski (Cray 1) 29 110503 33265 66530 1988-01-28 Walter N. Colquitt & Luther Welsch, Jr. 30 132049 39751 79502 1983-09-19 David Slowinski (Cray X-MP) 31 216091 65050 130100 1985-09-01 David Slowinski (Cray X-MP) 32 756839 227832 455663 1992-04-01 David Slowinski & Paul Gage (Cray 2) 33 859433 258716 517430 1994-02-01 David Slowinski & Paul Gage (Cray C90) 34 1257787 378632 757263 1996-09-03 David Slowinski & Paul Gage (Cray T94) 35 1398269 420921 841842 1996-11-13 Joel Armengaud, George Woltman & GIMPS (PC Pentium 90) 36 2976221 895932 1791864 1997-08-24 Gordon Spence, George Woltman & GIMPS (PC Pentium 100) 37 3021377 909526 1819050 1998-01-27 Roland Clarkson, Woltman, Scott Kurowski & GIMPS (Pentium 200) 38 6972593 2098960 4197919 1999-06-01 Nayan Hajratwala, Woltman, Kurowski & GIMPS (Pentium II 350) 39 13466917 4053946 8107892 2001-11-14 Michael Cameron & GIMPS (800 MHz AMD T-Bird PC) 40! 20996011 6320430 12640858 2003-11-17 Michael Shafer & GIMPS (2 GHz Pentium 4 Dell Dimension PC) 41! 24036583 7235733 14471465 2004-05-15 Josh Findley & GIMPS (2.4 GHz Pentium 4) 42! 25964951 7816230 15632458 2005-02-18 Nowak, Woltman, Kurowski, et al (2.4 GHz Pentium 4) 43? 30402457 9152052 18304103 2005-12-15 Cooper, Boone, Woltman, Kurowski, et al 44? 32582657 9808358 19616714 2006-09-04 Cooper, Boone, Woltman, Kurowski, et al 45? 37156667 11185272 22370543 2008-09-06 Hans-Michael Elvenich, George Woltman, Scott Kurowski, et al 46? 42643801 12837064 25674127 2009-04-12 Strindmo, Woltman, Kurowski, et al (GIMPS, PrimeNet) 47? 43112609 12978189 25956377 2008-08-23 Smith, Woltman, Kurowski, et al (GIMPS, PrimeNet) The ? and ! marks indicates that it is not yet known if there is an undiscovered Mersenne primes smaller than this one. The ! mark indicates that all primes < q have been tested at least once and the count is probably correct. Using my Extra Precision Integer Calculator XICalc: Command: q47 = 43112609 q47 = 431,12609 Command: M47 = 2^q47 - 1 M47 = 3164,70269,33025,59231,43453,72394,93375,16054,10618,84752,64644,14030,417 67,32811,24749,30693,68692,04318,51216,11837,8...,33048,24083,11909,31959,98014, 56245,63479,41202,19590,09280,79670,72944,79216,16491,88747,82657,80022,18116,66 971,52511 (12978189 Digits) Command: P47 = M47 * (M47 + 1) / 2 P47 = 50,07671,56849,82361,39058,66129,92965,99259,83999,56081,77422,50043,36486 ,14028,55831,89003,01470,24486,48047,12520,107...,16272,77971,50003,16751,57698, 60676,90811,20454,39188,65729,53988,61670,53938,33393,87226,95299,03827,90922,11 453,78816 (25956377 Digits) ```

The way to determine if 2^q - 1 is prime, given that q is an odd prime, is to use the Lucas-Lehmer test:

### ``` Lucas-Lehmer-Test(q): u := 4 for i := 3 to q do u := (u^2 - 2) mod (2^q - 1) enddo if u == 0 then 2^q - 1 is prime else 2^q - 1 is composite endif EndTest ```

Go to Plot of Mersenne Primes
Go to Plot of Mersenne Prime Discoveries