# Mersenne primes

Welcome to the Wikiversity Mersenne Primes Learning Project.
Marin Mersenne's name is widely known because of his interest in prime numbers that are one less than a power of two. Mersenne also did research in the area of the theory of music and musical instruments (see Wikipedia). Participants in this learning project are encouraged to join the distributed computing project that allows the computing power of personal computers to help search for Mersenne prime numbers (see Research Project, below).

## Mersenne prime numbers

In Mathematics, Mersenne prime numbers are those prime numbers which are one less than a power of two.
$Mn=2^{n}-1\,\$
if $n\,\$ is not a prime number, then $n\,\$ can be written as, $n=a\cdot b\,\$
$Mn=2^{n}-1=2^{ab}-1=(2^{a}-1)\cdot \left(1+2^{a}+2^{2a}+2^{3a}+\dots +2^{(b-1)a}\right)\,\$
    This implies, the Mersenne number is not a prime number if n is not prime.

## To find Mersenne primes

| $n\,\$ | $Mn\,\$ | Prime factors |
| 2 | 3 | First Mersenne prime |
| 3 | 7 | Second Mersenne prime |
| 5 | 31 | Third Mersenne prime |
| 7 | 127 | Fourth Mersenne prime |
| 11 | 2047 | 23, 89 |
| 13 | 8191 | Fifth Mersenne prime |
| 17 | 131071 | Sixth Mersenne prime |
| 19 | 524287 | Seventh Mersenne prime |
| 23 | 8388607 | 47, 178481 |
| 29 | 536870911 | 233, 1103, 2089 |
| 31 | 2147483647 | Eighth Mersenne prime |
| 37 | 137438953471 | 223, 616318177 |
| 41 | 2199023255551 | 13367, 164511353 |
| 43 | 8796093022207 | 431, 9719, 2099863 |
| 47 | 140737488355327 | 2351, 4513, 13264529 |
| 53 | 9007199254740991 | 6361, 69431, 20394401 |
| 59 | 576460752303423487 | 179951, 3203431780337 |
| 61 | 2305843009213693951 | Ninth Mersenne prime |
| 67 | 147573952589676412927 | 193707721, 761838257287 |
| 71 | 2361183241434822606847 | 228479, 48544121, 212885833 |
| 73 | 9444732965739290427391 | 439, 2298041, 9361973132609 |
| 79 | 604462909807314587353087 | 2687, 202029703, 1113491139767 |
| 83 | 9671406556917033397649407 | 167, 57912614113275649187721 |
| 89 | 618970019642690137449562111 | Tenth Mersenne prime |
| 97 | 158456325028528675187087900671 | 11447, 13842607235828485645766393 |
| 101 | 2535301200456458802993406410751 | 7432339208719, 341117531003194129 |
| 103 | 10141204801825835211973625643007 | 2550183799, 3976656429941438590393 |
| 107 | 162259276829213363391578010288127 | Eleventh Mersenne prime |
| 109 | 649037107316853453566312041152511 | 745988807, 870035986098720987332873 |
| 113 | 10384593717069655257060992658440191 | 3391, 23279, 65993, 1868569, 1066818132868207 |
| 127 | 170141183460469231731687303715884105727 | Twelfth Mersenne prime |
| 131 | 2722258935367507707706996859454145691647 | 263, 10350794431055162386718619237468234569 |
| 137 | 174224571863520493293247799005065324265471 | 32032215596496435569, 5439042183600204290159 |
| 139 | 696898287454081973172991196020261297061887 | 5625767248687, 123876132205208335762278423601 |
| 149 | 713623846352979940529142984724747568191373311 | 86656268566282183151, 8235109336690846723986161 |
| 151 | 2854495385411919762116571938898990272765493247 | 18121, 55871, 165799, 2332951, 7289088383388253664437433 |
| 157 | 182687704666362864775460604089535377456991567871 | 852133201, 60726444167, 1654058017289, 2134387368610417 |
| 163 | 11692013098647223345629478661730264157247460343807 | 150287, 704161, 110211473, 27669118297, 36230454570129675721 |
| 167 | 187072209578355573530071658587684226515959365500927 | 2349023, 79638304766856507377778616296087448490695649 |
| 173 | 11972621413014756705924586149611790497021399392059391 | 730753, 1505447, 70084436712553223, 155285743288572277679887 |
| 179 | 766247770432944429179173513575154591809369561091801087 | 359, 1433, 1489459109360039866456940197095433721664951999121 |
| 181 | 3064991081731777716716694054300618367237478244367204351 | 43441, 1164193, 7648337, 7923871097285295625344647665764672671 |
| 191 | 3138550867693340381917894711603833208051177722232017256447 | 383, 7068569257, 39940132241, 332584516519201, 87274497124602996457 |
| 193 | 12554203470773361527671578846415332832204710888928069025791 | 13821503, 61654440233248340616559, 14732265321145317331353282383 |
| 197 | 200867255532373784442745261542645325315275374222849104412671 | 7487, 26828803997912886929710867041891989490486893845712448833 |
| 199 | 803469022129495137770981046170581301261101496891396417650687 | 164504919713, 4884164093883941177660049098586324302977543600799 |
| 211 | 3291009114642412084309938365114701009965471731267159726697218047 | 15193, 60272956433838849161, 3593875704495823757388199894268773153439 |

## Research Project

Join the Great Internet Mersenne Prime Search. Describe your experience below.