# 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.

${\displaystyle Mn=2^{n}-1\,\ }$

if ${\displaystyle n\,\ }$ is not a prime number, then ${\displaystyle n\,\ }$ can be written as, ${\displaystyle n=a\cdot b\,\ }$

${\displaystyle Mn=2^{n}-1=2^{ab}-1=(2^{a}-1)\cdot \left(1+2^{a}+2^{2a}+2^{3a}+\dots +2^{(b-1)a}\right)\,\ }$

This implies, the Mersenne number is not a prime number if n is not prime.

## To find Mersenne primes

${\displaystyle n\,\ }$ ${\displaystyle Mn\,\ }$ 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.