xmersenne1_081831.gif

Mersenne Primes

  • 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 2n-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 2n-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 2n-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 Mn Digits in Mn
    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.