Mersenne primes: Difference between revisions
Appearance
Content deleted Content added
m removed Category:Mathematics; added Category:Prime numbers using HotCat |
|||
(20 intermediate revisions by 14 users not shown) | |||
Line 1: | Line 1: | ||
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 [[w:Marin Mersenne|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|Research Project]], below). |
|||
==Mersenne prime numbers== |
|||
In [[School:Mathematics|Mathematics]], Mersenne prime numbers are those [[prime numbers]] which are one less than a power of two. |
In [[School:Mathematics|Mathematics]], Mersenne prime numbers are those [[prime numbers]] which are one less than a power of two. |
||
<math>Mn=2^n - 1\,\ </math> |
<math>Mn=2^n - 1\,\ </math> |
||
if <math>n\,\ </math> is not a prime number then |
if <math>n\,\ </math> is not a prime number, then <math>n\,\ </math> can be written as, <math>n=a\cdot b\,\ </math> |
||
<math>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)\,\ </math> |
<math>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)\,\ </math> |
||
Line 24: | Line 29: | ||
|5 |
|5 |
||
|31 |
|31 |
||
|'''Third |
|'''Third Mersenne prime''' |
||
|- |
|- |
||
|7 |
|7 |
||
|127 |
|127 |
||
|''' |
|'''Fourth Mersenne prime''' |
||
|- |
|- |
||
|11 |
|11 |
||
Line 37: | Line 42: | ||
|8191 |
|8191 |
||
|'''Fifth Mersenne prime''' |
|'''Fifth Mersenne prime''' |
||
|- |
|||
|17 |
|17 |
||
|131071 |
|131071 |
||
Line 51: | Line 57: | ||
|29 |
|29 |
||
|536870911 |
|536870911 |
||
|233, 1103, 2089 |
|||
| |
|||
|- |
|- |
||
|31 |
|31 |
||
Line 59: | Line 65: | ||
|37 |
|37 |
||
|137438953471 |
|137438953471 |
||
|223, 616318177 |
|||
| |
|||
|- |
|- |
||
|41 |
|41 |
||
|2199023255551 |
|2199023255551 |
||
|13367, 164511353 |
|||
| |
|||
|- |
|- |
||
|43 |
|43 |
||
|8796093022207 |
|8796093022207 |
||
|431, 9719, 2099863 |
|||
| |
|||
|- |
|- |
||
|47 |
|47 |
||
|140737488355327 |
|140737488355327 |
||
|2351, 4513, 13264529 |
|||
| |
|||
|- |
|- |
||
|53 |
|53 |
||
|9007199254740991 |
|9007199254740991 |
||
|6361, 69431, 20394401 |
|||
| |
|||
|- |
|- |
||
|59 |
|59 |
||
|576460752303423487 |
|576460752303423487 |
||
|179951, 3203431780337 |
|||
| |
|||
|- |
|- |
||
|61 |
|61 |
||
Line 87: | Line 93: | ||
|67 |
|67 |
||
|147573952589676412927 |
|147573952589676412927 |
||
|193707721, 761838257287 |
|||
| |
|||
|- |
|- |
||
|71 |
|71 |
||
|2361183241434822606847 |
|2361183241434822606847 |
||
|228479, 48544121, 212885833 |
|||
| |
|||
|- |
|- |
||
|73 |
|73 |
||
|9444732965739290427391 |
|9444732965739290427391 |
||
|439, 2298041, 9361973132609 |
|||
| |
|||
|- |
|- |
||
|79 |
|79 |
||
|604462909807314587353087 |
|604462909807314587353087 |
||
|2687, 202029703, 1113491139767 |
|||
| |
|||
|- |
|- |
||
|83 |
|83 |
||
|9671406556917033397649407 |
|9671406556917033397649407 |
||
|167, 57912614113275649187721 |
|||
| |
|||
|- |
|- |
||
|89 |
|89 |
||
Line 111: | Line 117: | ||
|97 |
|97 |
||
|158456325028528675187087900671 |
|158456325028528675187087900671 |
||
|11447, 13842607235828485645766393 |
|||
| |
|||
|- |
|- |
||
|101 |
|101 |
||
|2535301200456458802993406410751 |
|2535301200456458802993406410751 |
||
|7432339208719, 341117531003194129 |
|||
| |
|||
|- |
|- |
||
|103 |
|103 |
||
|10141204801825835211973625643007 |
|10141204801825835211973625643007 |
||
|2550183799, 3976656429941438590393 |
|||
| |
|||
|- |
|- |
||
|107 |
|107 |
||
|162259276829213363391578010288127 |
|162259276829213363391578010288127 |
||
|'''Eleventh Mersenne prime''' |
|||
| |
|||
|- |
|- |
||
|109 |
|109 |
||
|649037107316853453566312041152511 |
|649037107316853453566312041152511 |
||
|745988807, 870035986098720987332873 |
|||
| |
|||
|- |
|- |
||
|113 |
|113 |
||
|10384593717069655257060992658440191 |
|10384593717069655257060992658440191 |
||
|3391, 23279, 65993, 1868569, 1066818132868207 |
|||
| |
|||
|- |
|- |
||
|127 |
|127 |
||
|170141183460469231731687303715884105727 |
|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 [[w:Great Internet Mersenne Prime Search|Great Internet Mersenne Prime Search]]. Describe your experience below. |
|||
==See also== |
|||
*[http://mersennewiki.org/index.php/Main_Page Mersenne wiki] |
|||
[[Category:Prime numbers]] |
Latest revision as of 07:40, 25 February 2024
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
[edit | edit source]In Mathematics, Mersenne prime numbers are those prime numbers which are one less than a power of two.
if is not a prime number, then can be written as,
- This implies, the Mersenne number is not a prime number if n is not prime.
To find Mersenne primes
[edit | edit source]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
[edit | edit source]Join the Great Internet Mersenne Prime Search. Describe your experience below.