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.

See also

• Mersenne wiki