## Mersenne and Fermatprimes field

Let NMF(b,p,n)= (b^(p^(n+1))-1)/(b^(p^n)-1)
These numbers contain under the single general form the main remarkable numbers of number theory.Indeed, we obtains :
- for b=2 and n=0 those are Mersenne's numbers.
- for b=+/-2 et p=2 those are Fermat's numbers.
- for b even and p=2 those are Generalized Fermat.
- for b >2 and n=0 those are repunits base b. These numbers can be considered as Generalized Mersenne, they can be prime only if p is prime.
- for b <=-2 and n=0 those are rep.base-1.  These numbers can also be considered as Generalized Mersenne, with the same property for p.
- for p=3 we obtain numbers of the form (b3^n-1)b3^n+1 which we can name Generalized Mersenne-Fermat in particular for the combined form of their divisors 2k3n+1+1.

A general form of the divisors of these numbers will be noted of more : 2kpn+1+1 . We find again 2kp+1 for Mersenne's numbers and k2n+2+1 for Fermat's numbers.

For n=1 and p=m not necessarily prime we find also (bm+/-1)bm+1 (see New forms of primes), but the form of the divisors is more complex generally.

The table below summarizes the organization and the specificities of these numbers :

 Divisors Generalized Mersenne  (rep. base-1) Mersenne Generalized Mersenne  (repunits) n=0 2kp+1 ...... b=-6 b=-5 b=-4 b=-3 b=-2 b=2 b=3 b=4 b=5 b=6 ...... ...... (6p+1)/7 (5p+1)/6 (4p+1)/5 (3p+1)/4 (2p+1)/3 2p-1 (3p-1)/2 (4p-1)/3 (5p-1)/4 (6p-1)/5 ...... Generalized  Fermat Fermat Fermat Generalized  Fermat p=2 k2n+2+1 ...... b=-6 b=-5 b=-4 b=-3 b=-2 b=2 b=3 b=4 b=5 b=6 ...... ...... 62^n+1 =0 mod 2 42^n+1 =0 mod 2 22^n+1 22^n+1 =0 mod 2 42^n+1 =0 mod 2 62^n+1 ...... Generalized Mersenne-Fermat Generalized Mersenne-Fermat p=3 2k3n+1+1 ...... b=-6 b=-5 b=-4 b=-3 b=-2 b=2 b=3 b=4 b=5 b=6 ...... ...... (63^n-1)63^n+1 =0 mod 3 (43^n-1)43^n+1 (33^n-1)33^n+1 =0 mod 3 (23^n+1)23^n+1 (33^n+1)33^n+1 =0 mod 3 (53^n+1)53^n+1 (63^n+1)63^n+1 ...... p 2kpn+1+1 ...... ...... ...... ......

 Mersenne field primes or prps for p p < ....... ... ... (104^p-1)/103 97,263,5437 5438 ....... ... ... (99^p-1)/98 3,5,37,47,383,5563 5564 ....... ... ... (95^p-1)/94 7,523,9283,10487,11483 11484 ....... ... ... (90^p-1)/89 3,19,97,5209 5210 (89^p-1)/88 3,7,43,47,71,109,571,11071 11972 ....... ... ... (82^p-1)/81 2,23,31,41,7607 7608 ....... ... ... (70^p-1)/69 2,29,59,541,761,1013,11621 11622 (69^p-1)/68 3,61,2371,3557,8293 8294 ....... ... ... (67^p-1)/66 19,367,1487,3347,4451,10391 10392 ....... ... ... (62^p-1)/61 3,5,17,47,163,173,757,4567,9221,10889 10890 ....... ... ... (59^p-1)/58 3,13,479,12251 12252 ....... ... ... (50^p-1)/49 3,5,127,139,347,661,2203,6521 20000 (49^p-1)/48 - (48^p-1)/47 19,269,349,383,1303,15031 20000 (47^p-1)/46 127,18013 20000 (46^p-1)/45 2,7,19,67,211,433,2437,2719,19531 20000 (45^p-1)/44 19,53,167,3319,11257 20000 (44^p-1)/43 5,31,167 20000 (43^p-1)/42 5,13,6277 20000 (42^p-1)/41 2,1319 20000 (41^p-1)/40 3,83,269,409,1759,11731 20000 (40^p-1)/39 2,5,7,19,23,29,541,751,1277 20000 (39^p-1)/38 349,631,4493,16633 20000 (38^p-1)/37 3,7,401,449 20000 (37^p-1)/36 13,71,181,251,463,521,7321 20000 (36^p-1)/35 2 - (35^p-1)/34 313,1297 20000 (34^p-1)/33 13,1493,5851,6379 20000 (33^p-1)/32 3,197,3581,6871 20000 (32^p-1)/31 20000 (31^p-1)/30 7,17,31,5581,9973 20000 (30^p-1)/29 2,5,11,163,569,1789,8447 20000 (29^p-1)/28 5,151,3719 20000 (28^p-1)/27 2,5,17,457,1423 20000 (27^p-1)/26 3 - (26^p-1)/25 7,43,347,12421,12473,26717 30000 (25^p-1)/24 - (24^p-1)/23 3,5,19,53,71,653,661,10343 30000 (23^p-1)/22 5,3181 30000 (22^p-1)/21 2,5,79,101,359,857,4463,9029,27823 30000 (21^p-1)/20 3,11,17,43,271 30000 (20^p-1)/19 3,11,17,1487 30000 (19p-1)/18 19,31,47,59,61,107,337,1061,9511,22051 30000 (18p-1)/17 2,25667,28807 30000 (17p-1)/16 3,5,7,11,47,71,419,4799 30000 (16p-1)/15 2 - (15p-1)/14 3,43,73,487,2579,8741 30000 (14p-1)/13 3,7,19,31,41,2687,19697,..,59693,67421 30000 (13p-1)/12 5,7,137,283,883,991,1021,1193,3671,18743,31751 31752 (12p-1)/11 2,3,5,19,97,109,317,353,701,9739,14951,37573,46889 46890 (11p-1)/10 17,19,73,139,907,1907,2029,4801,5153,10867,20161 41000 (10p-1)/9 2,19,23,317,1031,49081,86453,109297,270343 300000 (9p-1)/8 - (8p-1)/7 3 - (7p-1)/6 5,13,131,149,1699,14221,35201,126037 126038 (6p-1)/5 2,3,7,29,71,127,271,509,1049,6389,6883,10613,19889,...,79987 50000 (5p-1)/4 3,7,11,13,47,127,149,181,619,929,3407,10949,13241,13873,16519 60000 (4p-1)/3 2 - (3p-1)/2 3,7,13,71,103,541,1091,1367,1627,4177,9011,9551,36913,43063,49681, 57917 90000 2p-1 2,3,5,7,13,17,19,31,61,89,107,127,521,607,1279,2203,2281,3217,4253, 4423,9689,9941,11213,19937,21701,23209,44497,86243,110503,132049, 216091,756839,859433,1257787,1398269,2976221,3021377,6972593, 13466917,..,20996011,..,24036583,..,25964951,..,30402457,..,32582657, ..,37156667,..,43112609 18816700 (2p+1)/3 3,5,7,11,13,17,19,23,31,43,61,79,101,127,167,191,199,313,347,701,1709, 2617,3539,5807,10501,10691,11279,12391,14479,42737,83339,95369, 117239,127031,138937,141079,267017,269987,374321,..,986191 720000 (3p+1)/4 3,5,7,13,23,43,281,359,487,577,1579,1663,1741,3191,9209,11257,12743, 13093,17027,26633,...,104243,...,134227 85000 (4p+1)/5 3 - (5p+1)/6 5,67,101,103,229,347,4013,23297,30133 65000 (6p+1)/7 3,11,31,43,47,59,107,811,2819,4817,9601,33581,38447,41341 55000 (7p+1)/8 3,17,23,29,47,61,1619,18251 55000 (8p+1)/9 - (9p+1)/10 3,59,223,547,773,1009,1823,3803,49223 49224 (10p+1)/11 5,7,19,31,53,67,293,641,2137,3011 43000 (11p+1)/12 5,7,179,229,439,557,6113 40000 (12p+1)/13 5,11,109,193,1483,11353,21419,21911,24071 40000 (13p+1)/14 3,11,17,19,919,1151,2791,9323 40000 (14p+1)/15 7,53,503,1229,22637 30000 (15p+1)/16 3,7,29,1091,2423 30000 (16p+1)/17 3,5,7,23,37,89,149,173,251,307,317,30197 40000 (17p+1)/18 7,17,23,47,967,6653,8297 30000 (18p+1)/19 3,7,23,73,733,941,1097,1933,4651 30000 (19p+1)/20 17,37,157,163,631,7351,26183 30000 (20^p+1)/21 2,5,79,89,709,797,1163,6971 30000 (21^p+1)/22 3,5,7,13,37,347,17597 30000 (22^p+1)/23 3,5,13,43,79,101,107,227,353,7393 30000 (23^p+1)/24 11,13,67,109,331,587 20000 (24^p+1)/25 2,7,11,19,2207,2477,4951 20000 (25^p+1)/26 3,7,23,29,59,1249,1709,1823,1931,3433,8863 20000 (26^p+1)/27 11,109,227,277,347,857,2297,9043 20000 (27^p+1)/28 - (28^p+1)/29 3,19,373,419,491,1031 20000 (29^p+1)/30 7 20000 (30^p+1)/31 2,139,173,547,829,2087,2719,3109,10159 20000 (31^p+1)/32 109,461,1061 20000 (32^p+1)/33 2 20000 (33^p+1)/34 5,67,157,12211 20000 (34^p+1)/35 3 20000 (35^p+1)/36 11,13,79,127,503,617,709,857,1499,3823 20000 (36^p+1)/37 31,191,257,367,3061 20000 (37^p+1)/38 5,7,2707 20000 (38^p+1)/39 2,5,167,1063,1597,2749,3373,13691 20000 (39^p+1)/40 3,13,149,15377 20000 (40^p+1)/41 53,67,1217,5867,6143,11681 20000 (41^p+1)/42 17,691 20000 (42^p+1)/43 2,3,709,1637,17911 20000 (43^p+1)/44 5,7,19,251,277,383,503,3019,4517,9967 20000 (44^p+1)/45 2,7 20000 (45^p+1)/46 103,157 20000 (46^p+1)/47 7,23,59,71,107,223,331,2207,6841 20000 (47^p+1)/48 5,19,23,79,1783,7681 20000 (48^p+1)/49 2,5,17,131 20000 (49^p+1)/50 7,19,37,83,1481,12527 20000 (50^p+1)/51 1153 20000 ...... ... ... (58^p+1)/59 3,17,1447,11003 11004 ...... ... ... (94^p+1)/95 71,307,613,1787,3793,10391 10392 ...... ... ... (100^p+1)/101 3,293,461,11867 11868 ...... ... ... (256p+1)/257 5,13 7000 ...... ... ... (1296^p+1)/1297 3,2153,3517 3518 ...... ... ... (65536p+1)/65537 239 7000 ...... ... ...

You can also consult the next links :

Created by  Henri Lifchitz : April, 12 1999, last modification: September, 21 2008.
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}