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Revision difference : table of factors of small Mersenne numbers |
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The following table lists the prime factors for some small Mersenne numbers, specifically, numbers of the form $2^p - 1$ where $p$ is a prime in the range $1 < p < 200$ and $2^p - 1$ is composite. Thus, the following values of $p$ have been left off the table: 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127 and all primes greater than 199 (see table of Mersenne primes). The numbers get quite large, and correspondingly, so do their factors. All these numbers are squarefree (given the M\"obius function, $\mu(2^p - 1) \neq 0$). Their factors are all of the form $2mp + 1$, with $m$ in the range $0 < m < 2^{p - 1}$.
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The following table lists the prime factors for some small Mersenne numbers, specifically, numbers of the form $2^p - 1$ where $p$ is a prime in the range $1 < p < 100$ and $2^p - 1$ is composite. Thus, the following values of $p$ have been left off the table: 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, and all primes greater than 97 (see table of Mersenne primes). The numbers get quite large, and correspondingly, so do their factors. All these numbers are squarefree (given the M\"obius function, $\mu(2^p - 1) \neq 0$). Their factors are all of the form $2mp + 1$, with $m$ in the range $0 < m < 2^{p - 1}$.
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| \begin{tabular}{|r|l|} |
\begin{tabular}{|r|l|} |
| $p$ & Factors of $2^p - 1$ \\ |
$p$ & Factors of $2^p - 1$ \\ |
| 11 & 23, 89 \\ |
11 & 23, 89 \\ |
| 23 & 47, 178481 \\ |
23 & 47, 178481 \\ |
| 29 & 233, 1103, 2089 \\ |
29 & 233, 1103, 2089 \\ |
| 37 & 223, 616318177 \\ |
37 & 223, 616318177 \\ |
| 41 & 13367, 164511353 \\ |
41 & 13367, 164511353 \\ |
| 43 & 431, 9719, 2099863 \\ |
43 & 431, 9719, 2099863 \\ |
| 47 & 2351, 4513, 13264529 \\ |
47 & 2351, 4513, 13264529 \\ |
| 53 & 6361, 69431, 20394401 \\ |
53 & 6361, 69431, 20394401 \\ |
| 59 & 179951, 3203431780337 \\ |
59 & 179951, 3203431780337 \\ |
| 67 & 193707721, 761838257287 \\ |
67 & 193707721, 761838257287 \\ |
| 71 & 228479, 48544121, 212885833 \\ |
71 & 228479, 48544121, 212885833 \\ |
| 73 & 439, 2298041, 9361973132609 \\ |
73 & 439, 2298041, 9361973132609 \\ |
| 79 & 2687, 202029703, 1113491139767 \\ |
79 & 2687, 202029703, 1113491139767 \\ |
| 83 & 167, 57912614113275649087721 \\ |
83 & 167, 57912614113275649087721 \\ |
| 97 & 11447, 13842607235828485645766393 \\ |
97 & 11447, 13842607235828485645766393 \\ |
| 101 & 7432339208719, 341117531003194129 \\ |
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| 103 & 2550183799, 3976656429941438590393 \\ |
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| 109 & 745988807, 870035986098720987332873 \\ |
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| 113 & 3391, 23279, 65993, 1868569, 1066818132868207 \\ |
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| 131 & 263, 10350794431055162386718619237468234569 \\ |
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| 137 & 32032215596496435569, 5439042183600204290159 \\ |
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| 139 & 5625767248687, 123876132205208335762278423601 \\ |
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| 149 & 86656268566282183151, 8235109336690846723986161 \\ |
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| 151 & 18121, 55871, 165799, 2332951, 7289088383388253664437433 \\ |
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| 157 & 852133201, 60726444167, 1654058017289, 2134387368610417 \\ |
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| 163 & 150287, 704161, 110211473, 27669118297, 36230454570129675721 \\ |
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| 167 & 2349023, 79638304766856507377778616296087448490695649 \\ |
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| 173 & 730753, 1505447, 70084436712553223, 155285743288572277679887 \\ |
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| 179 & 359, 1433, 1489459109360039866456940197095433721664951999121 \\ |
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| 181 & 43441, 1164193, 7648337, 7923871097285295625344647665764672671 \\ |
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| 191 & 383, 7068569257, 39940132241, 332584516519201, 87274497124602996457 \\ |
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| 193 & 13821503, 61654440233248340616559, 14732265321145317331353282383 \\ |
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| 197 & 7487, 26828803997912886929710867041891989490486893845712448833 \\ |
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| 199 & 164504919713, 4884164093883941177660049098586324302977543600799 \\ |
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| \end{tabular} |
\end{tabular} |
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