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

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This page accessed times since October 20, 2004.
Page created by: hjsmithh@sbcglobal.net
Changes last made on Tuesday, 07-Jul-09 12:59:40 PDT