table of Mersenne primes

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

Canonical name

TableOfMersennePrimes

Date of creation

2013-03-22 18:04:25

Last modified on

2013-03-22 18:04:25

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

7

Author

PrimeFan (13766)

Entry type

Data Structure

Classification

msc 11A41

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