# table of Mersenne primes

This is a table of the known Mersenne primes. This table could be complete, but it could just as easily be hopelessly short of completeness.

The first few Mersenne primes are so small written in base 10 that there is no excuse not to do so. Furthermore, since these were known since antiquity and the name of the first discoverer can be neither ascertainted nor disputed, we can dispense with the “Discoverer” field and instead use it for the associated perfect number (or 2-perfect number, to be more precise, see: multiply perfect number). The first field gives the rank (the Mersenne prime’s position in A000396 of Sloane’s OEIS), the second field gives the exponent $n$ (for the formula $2^{n}-1$), the third field gives the Mersenne prime written in base 10, and the last field gives the associated 2-perfect number.

Rank Exponent Prime 2-Perfect
1 2 3 6
2 3 7 28
3 5 31 496
4 7 127 8128
5 13 8191 33550336

The number 1 has been left off this listing, not out of some dogmatic belief that it is not a prime number, but because accepting it as a Mersenne prime one would have to also explain the fact that it leads to a 1-perfect number instead of a 2-perfect number like the other Mersenne primes. So for this particular application, 1 can’t be considered a Mersenne prime.

Moving along, the associated 2-perfect numbers are getting too large, so we’ll omit them for the rest of the table, and reinstate the “Discoverer” field.

Rank Exp. Prime Discoverer
6 17 131071 Pietro Cataldi, 1588
7 19 524287 Pietro Cataldi, 1588
8 31 2147483647 Leonhard Euler, 1772
9 61 2305843009213693951 Ivan Pervushin, 1883
10 89 618970019642690137449562111 R. E. Powers, 1911
11 107 162259276829213363391578010288127 R. E. Powers, 1914
12 127 170141183460469231731687303715884105727 Édouard Lucas, 1876

Now it starts to become impractical to write out these numbers in base 10. So we’ll settle for scientific notation, with our third field giving a number between 0 and 10, our fourth field giving the exponent to which to raise 10 and multiply by the number in the third field, and the name of the discoverer is moved to the fifth field. The number in scientific notation is of course useless if you wish to perform any kind of modular arithmetic, but it can be helpful in order to computer other things, such as how many football fields it would take to write the number out in base 10 in 12 point Courier New. If you just want to know how many base 10 digits it has, just add 1 to the scientific notation exponent. By contrast, the bits (binary digits) of these Mersenne primes have an extremely efficient, lossless run-length encoding: the exponent indicates how many 1s the number consists of in binary.

Rank Exp. Prime (SCN) SCN exp. Discoverer
13 521 6.864797660130609 156 Robinson, 1952
14 607 5.311379928167670 182 Robinson, 1952
15 1279 1.040793219466439 385 Robinson, 1952
16 2203 1.475979915214180 663 Robinson, 1952
17 2281 4.460875571837584 686 Robinson, 1952
18 3217 2.591170860132026 968 Riesel, 1957
19 4253 1.907970075244390 1280 Alexander Hurwitz, 1961
20 4423 2.855425422282796 1331 Alexander Hurwitz, 1961
21 9689 4.782202788054612 2916 Donald B. Gillies, 1963
22 9941 3.460882824908512 3375 Donald B. Gillies, 1963
23 11213 2.814112013697373 6001 Donald B. Gillies, 1963
24 19937 4.315424797388162 6532 Bryant Tuckerman, 1971
25 21701 4.486791661190433 6532 Nickel, L. Curt Noll, 1978
26 23209 4.028741157789888 6986 L. Curt Noll, 1979
27 44497 8.545098243036338 13394 David Slowinski, Nelson, 1979
28 86243 5.369279955027563 25961 David Slowinski, 1982
29 110503 5.219283133417550 33264 Welsh, Colquitt, 1988
30 132049 5.127402762693207 39750 David Slowinski, 1983
31 216091 7.460931030646613 65049 David Slowinski, 1985
32 756839 1.741359068200870 227831 David Slowinski, Paul Gage, 1992
33 859433 1.294981256042076 258715 David Slowinski, Paul Gage, 1994
34 1257787 4.122457736214286 378631 David Slowinski, Paul Gage, 1996
35 1398269 8.147175644125730 420920 Great Internet Mersenne Prime Search (GIMPS), Joel Armengaud, George F. Woltman, 1996
36 2976221 6.233400762485786 895931 GIMPS, Gordon Spence, George F. Woltman, 1997
37 3021377 1.274116830300933 909525 GIMPS, Roland Clarkson, George F. Woltman, Scott Kurowski, 1998
38 6972593 4.370757441270813 2098959 GIMPS, Najan Hajratwala, George F. Woltman, Scott Kurowski, 1999

These next few Mersenne primes have question marks next to their rank. If any other Mersenne primes are found with exponents between 13466917 and 32582657, the ranks of at least one of these will have to be adjusted. But if no others are found in that range, the question marks can simply be removed.

 Rank Exp. Prime (SCN) SCN exp. Discoverer 39? 13466917 9.249477380067013222477584 4053945 GIMPS, Michael Cameron, George F. Woltman, Scott Kurowski, 2001 40? 20996011 1.259768954503301050204943 6320429 GIMPS, Michael Shafer, George F. Woltman, Scott Kurowski, 2003 41? 24036583 2.994104294041571720890489 7235732 GIMPS, Josh Findley, George F. Woltman, Scott Kurowski, 2004 42? 25964951 1.221646300612779481077540 7816229 GIMPS, Dr. Martin Novak MD, George F. Woltman, Scott Kurowski, 2005 43? 30402457 3.154164756188460809363030 9152051 GIMPS, Curtis Cooper, Steven R. Boone, George F. Woltman, Scott Kurowski, 2005 44? 32582657 1.245750260153694554008555 9808357 GIMPS, Curtis Cooper, Steven R. Boone, George F. Woltman, Scott Kurowski, 2006
Title table of Mersenne primes TableOfMersennePrimes 2013-03-22 18:04:25 2013-03-22 18:04:25 PrimeFan (13766) PrimeFan (13766) 7 PrimeFan (13766) Data Structure msc 11A41