Name for Father Marin Mersenne (1588-1648) who in his time provided the service to the mathematical community of communicating many of the mathematical discoveries of the day to others. Without a lot of communication tools available this was sometimes the only way the work every got published or shared. Mersenne was a great contributor of this service publishing a lot of work of the day including a lot of Fermat's work.

Mersenne mentioned in one of his letters the possibility of prime numbers that were in the form 2^{n}-1 and then came up with a interesting list of numbers n that it was prime for. The list he came up with was n=2,3,5,7,13,17,19,31,67,127,257 and made the claim that there was no other n below 257.

Since that time there have been corrections to some of the mistakes that were found in his original writings. His basic premince held fast though and today there is more imformation about the primes that now bear his name.

One of which was that in order for the term 2^{n}-1 to be prime the number n itself must also be prime. There were also numbers discovered that were prime values of n that did not produce a prime number. For example if we let n=11 we get that 2^{n}-1=2047. However, 2047 is the product of the numbers 23 and 89 so the number is obviously not prime.

Today there are 43 known Mersenne primes. There were some corrections made to his original list and today's current list is given below:

#

n

M_{n}

Digits in M_{n}

1

2

3

1

2

3

7

1

3

5

31

2

4

7

127

3

5

13

8191

4

6

17

131071

6

7

19

524287

6

8

31

2147483647

10

9

61

2305843009213693951

19

10

89

618970019…449562111

27

11

107

162259276…010288127

33

12

127

170141183…884105727

39

13

521

686479766…115057151

157

14

607

531137992…031728127

183

15

1,279

104079321…168729087

386

16

2,203

147597991…697771007

664

17

2,281

446087557…132836351

687

18

3,217

259117086…909315071

969

19

4,253

190797007…350484991

1,281

20

4,423

285542542…608580607

1,332

21

9,689

478220278…225754111

2,917

22

9,941

346088282…789463551

2,993

23

11,213

281411201…696392191

3,376

24

19,937

431542479…968041471

6,002

25

21,701

448679166…511882751

6,533

26

23,209

402874115…779264511

6,987

27

44,497

854509824…011228671

13,395

28

86,243

536927995…433438207

25,962

29

110,503

521928313…465515007

33,265

30

132,049

512740276…730061311

39,751

31

216,091

746093103…815528447

65,050

32

756,839

174135906…544677887

227,832

33

859,433

129498125…500142591

258,716

34

1,257,787

412245773…089366527

378,632

35

1,398,269

814717564…451315711

420,921

36

2,976,221

623340076…729201151

895,932

37

3,021,377

127411683…024694271

909,526

38

6,972,593

437075744…924193791

2,098,960

39

13,466,917

924947738…256259071

4,053,946

40

20,996,011

125976895…855682047

6,320,430

41

24,036,583

299410429…733969407

7,235,733

42

25,964,951

122164630…577077247

7,816,230

43

30,402,457

315416475…652943871

9,152,052

Today the search continues for more Mersenne primes and at the same time the largest known primes. One of the greatest resources in this search is the Great Internet Mersenne Prime Search (GIMPS). This is a small program designed to be downloaded on personal PCs so the total results can be compiled togeather. This saves the search from having to be run on supercomputers. Similar programs are used for the SETI project.

I am currently running 2 computers (soon hopefully more) that are running the search for new primes and verifiying current data. If you're intersted in contributing or want to see what my computer does (you won't see much) let me know or visit the website.