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Mersenne and Fermat primes 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
......

...

...


  

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Created by  Henri Lifchitz : April, 12 1999, last modification: September, 21 2008.