New forms of primes



Let the numbers of the following forms:

The primes of the forms (2 n - k)*2n + 1 (for all k<2n) and (2 n +k)*2n - 1 (for even k<2n) can be tested with Proth's Theorem (N-1 test),  the forms  (2 n - k)*2n - 1 (for all k<2n) and (2 n +k)*2n - 1 (for even k<2n) with Lucas  Theorem (N+1 test), the others forms seem more difficult to test.

So we can obtain primes as less as big as the others forms k*2n+1 ou k*2n-1 (see Top below).



These numbers present behaviour resemblances with the forms k*2n+1 ou k*2n-1:

If we puts Np(k,n)=k*2n+1 and Nm(k,n)=k*2n-1 there exists many relationships and arithmetical properties with the new primes. Relationships with Mersenne numbers are as evident : For example : Nmp(2,n)=Mn2.



First numerical results :
            subscripts :  t = twin, s = Sophie Germain, c = Cunningham chain of second kind, b = BiTwin.
 
k n,   Nmp(k,n)=(2 n - k)*2n + 1   prime n max
1 1,2t,4t,32 400000(a)
2 Nmp(2,n)=(2n-1)2=Mn2 -
3 3s,5,7s,9,11,13s,15,19s,35,39,45,51,59,213,607,1315,1435 3000
4 4t,8,12,14,18,32,62tc,66,96,120,122c,140t,180,202,228,740,800,810,1012,1142,1386, 
1496,1698,4622,5674
19000
5 nkS(3,5) -
6 3 4000
7 ? 180000 (c)
8 5,7t,13,15t,27,57,163,217,245,253,637,753,1605,4607,4703,5575,5843,7013, 
8913,12645
12689
9 4,7,19,43,46,82,199,247,266,310,379,857 1000
10 4c,6,10,12,14c,22,30,34,72,172,264,322,430 1000
11 5,11,27,77,99,285 1000
12 5,6,11,14,21,45,106,111,114,166,221,230,275 1000
13
6564
90000 (c)
14 5,7t,9,11t,13,19,25,73,209,361,381,409,471,905 1000
15 4,6,14,22,50,238,258,678 1000
16 8,10t,16,28,34,62,70,160,226,346,440,446,524,1138,1264,1984,2096,3370,5236,5686, 
5918,15440,19592
19681
17 13,15,21,37,39,71,205,227,231,793 1000
18 5,7,13,23,25,35,77,875 1000
19 12,30,44,206,266,340,436,628,682 1000
20 nkS(3,5) -
21 5,6s,7,8,9s,11,13s,14,15,20,28,45,49,50,57,92,100,101,175,188,428,434,459,478,498, 
638,680,731,883,1032,1126,2362,2667,3520,3854,4004,5116,6051,6295,7659,8796, 
8862
14000
22 6,42,54,254,506 1000
23 7,17,89,103 1000
24 5,7,8,9,14,359,394 1000
25 10,12,18,30,42,90,282 1000
26 5,11,19,109,445,469,739,881 1000
27 15,21,69,141,309,591,3165 5000
28 8,64,280,608 1000
29 5,45,375,393 1000
30 10,14,22,34,58,122,722 1000
31 6t,8c,12t,18,20,40,160,228,244,260,574,736,820 1000
32 7,17,29,35,37,159,241,277,317,327,395,1181,1675,1763,2043,2083,2099,3973,4055, 
6059,6605,6771
8567
33 7,8,16,21,41,48,91,92,148,187,424,527,532,708,833 1000
34 ? 100000 (c)
35 nkS(3,5) -
36 7,11,13,18,31,42,43,141,147,230,239,255,267,295,651 1000
37 26,50,850 1000
38 9,15,37,39,59,105,877 1000
39 6,7,12,18,92,119,236,551,569,637 1000
40 20,24,140,276,390 1000
41 57,327 1000
42 6,9,39,46,50,57,67,82,87,206,282,309,521 1000
43 16,44 1000
44 7,11,49,61,79,139,167,173,269,301 1000
45 6,12,16,18,24,32,72,88,232,250,312,696,878 1000
46 6,12,16,24,42,84,106,114,274 1000
47 7,9,13,19,31,35,37,47,73,79,113,215,299,559 1000
48 12,36,468 1000
49 18,22,70,90,110,192,288,318,658 1000
50 nkS(3,5) -
51 7,19,25,53,187,251 1000
52 6,10,86,286,434,482,694 1000
53 7,9,11,21,69,153,157,193,219,229,237,367,479,537,677,697 1000
54 6s,7s,8,14,17,18s,20s,21,27,32s,37,40,42,53,80,89,92,107,109,215,225,469,474,559,997, 
1847
3200
55 6,120 1000
56 11,21,45,51,59,85,93,157,159,225 1000
57 6,9,22,42,54,57,79,153,297 1000
58 20,32,52,112,148 1000
59 95,121,275 1000
60 6,8,18,172,190,334 1000
61 6,8,10,14,24,34,36,50,62,84,130,178,534,930 1000
62 27c,51,93,189,747c 1000
63 8,12,33,37,144,207,328,348,537,635 1000
64 12,22,102t,114,426 1000
65 nkS(3,5) -
66 7,9,17,91,137,213 1000
67 10,14,38,46,74,122 1000
68 7c,13,15,25,27,37,43,47,51,93,177,525,575,597,789 1000
69 9,12,30,69,138,267,429,621 1000
70 12,18,20c,34,62,82,146,226,238,410,444,628,714,942 1000
...    
79
?
150000 (c)

  (a) with help Didier Boivin       (c) with help of Donovan Johnson
 


k n,  Nmm(k,n)=(2 n - k)*2n - 1  prime n max
1 2t,4t,5,9,10,18,38,45,50,57,108,161,208,224,225,240,354,597,634,1008,1080,1468,1525,1560 3000 (b)
2 2,3,4,6,7,10,12,15,18,19,21,25,27,55,129,132,159,171,175,315,324,358,393,435,786,1459,1707,2923 5000 (b)
3 2,nkB(3) -
4 3,4t,10,35,47,62t,71,82,140t,335,484,502,655,1451,1475,1934,2464,2647,3562 5000 (b)
5 3,5,9,101,1193 14000 (b)
6 nkB(3) -
7 3,8,17,49,115,227,416,478,977,1052,1652,1702 3000 (b)
8 4,6,7t,8,12,15t,20,30,54,72,84,110,126,139,150,191,211,285,303,369,376,392,447,1566,1588,1795,2550
2743
4000 (b)
9 nkB(3) -
10 15,31,103,135,253,1141 3000 (b)
11 4,6,13,14,22,28,34,36,41,69,76,93,141,209,216,281,558,617,648,917,1229,1472 2000 (b)
12 nkB(3) -
13 4,5,8,9,10,14,20,26,28,31,33,41,49,57,92,98,131,344,371,472,609,1136,1589,1853 2000 (b)
14 4,7t,10,11t,20,23,26,31,46,52,56,98,547,1256 2000 (b)
15 nkB(3) -
16 9,10,45,184,325,604,642,994,1368,1756 3000 (d)
...    
32 9,11,24,59,79,90,113,125,163,186,199,239,311,312,350,434,543,1833,2022,3422,4624,4812,5445,5856 7000 (d)
34 11,18,26,35,95,98,171,211,374,379,391,815,915,950,2930,4846,5127,7319,7835,9586 10000 (a)
...    
64 7,40,47,62,75,80,87,102,112,119,135,219,295,440 2000 (d)
...    
127 7,8,11,13,16,17,53,59,88,97,102,123,134,149,155,212,253,278,795,1157,1395,2941 4000 (d)
128 10,16,1735,2235 4000 (d)
...    
256 9,10,16,22,38,44,69,109,200,205,440,514,704,1056,1576 3000 (d)
...    
511 13,17,29,32,94,326,385,970,992,1018,1186,1232,2470,3994,4717,5258,6514 8000 (d)
512 10,12,14,16,24,27,30,33,36,49,63,94,96,100,137,160,190,294,338,376,387,418,825,1548,1657,1824,1940,3341,3485,5604,7077,7433,8753,9495,22770,26484,29757 30000 (d)
...    
1024 31,118,124,138,279,310,315,318,696,747,1176 3000 (d)

(a) by Didier Boivin     (b) with help of Gary Chaffey      (d) by  Gary Chaffey

 
 

k n,  Npm(k,n)=(2 n +k)*2n - 1  prime n max
1 1t,2,3t,4,6,10,16,24,26,35,52,55,95,144,379,484,939 1000
2 1,2,3,5,8,9,12,15,17,18,21,23,27,32,51,65,87,180,242,467,491,501,507,555,680,800 1000
3 nkB(3) -
4 1,2,5,13,16,20,26,118,128,209,269,296 1000
5 1,3,79 1000
6 nkB(3) -
7 1t,2,4,13,40,88,157,652,782,964 1000
8 1,2,3,4,5,9,18,20,21,29,33,61,63,65,104,135,137,204,282,978 1000
9 nkB(3) -
10 1,299,569 1000
11 2t,3,4t,6t,7,8,23,24,36,52,58,192,224,336,398,544,631 1000
12 nkB(3) -
13 1t,2,3,4,5,7t,8,11,15,22,23,25,27,40,49,62,74,151,187,385,467,650 1000
14 1,2t,4,5,8,10t,13,20,40,104,110,170,377 1000
15 nkB(3) -

 
 

k n,  Npp(k,n)=(2 n + k)*2n + 1  prime n max
1 1,3,9 40000
2 Npp(2,n)=(2n+1)2 -
3 1,2,3,6,10,17,37,70,105,126,155,165,215,765 1000
4 1t,3,5t,15,17,21,159,161 1000
5 2,4,10,16,20,22,56,68,128,410 1000
6 1,2s,3s,4,5,6s,8,10,11s,12s,19,27,28,32,36,48,56,61,131,170,251,750,771,779,790,874, 
1176,1728,2604
3000
7 1t,5,9,11,15,81 1000
8 30,498 1000
9 1,2,3,4,6,14,22,24,68,86,279,645,776,959 1000
10 nkS(3,5) -
11 2t,4t,6t,16,22,28,34,234,452,476,554,992 1000
12 1s,4,5s,7,17,49,67,68,115,445 1000
13 1t,7t,43,67,107,203,953 1000
14 2t,6,10t,14,22,50,158,418,562,570 1000
15 ? 100000 (c)
16 1,3t,7,9,13,15,19,25,63,75,85,113,119,125,199,363,437,815,885 1000
17 32,68,528 1000
18 1s,2s,5s,10s,14,29,55,85s,410 1000
19 1,595 1000
20 2c,4c,8,18,22,28,36,38,68,82,124,202,356,452,568,802 1000
21 1,2,3,4,5,6,7,8,10,12,13,14,15,21,23,36,48,51,52,181,212,338,651,764,895,943 1000
22 3t,39,423 1000
23 2,6,10,82,270,870 1000
24 1s,2s,3,4s,7,10s,16,17,23,27,43,49,126s,187,220,257,443,577 1000
25 nkS(3,5) -

(c) with help of Donovan Johnson


New "Cullen" and "Woodall" :

NCmp(n)=(2 n - n)*2n + 1 prime for n =1,3s,4t,10,11,16,47,57,69,166,327,460,1108,4740,20760,21143  (n<=22000)
NWmm(n)=(2 n - n)*2n - 1 prime for n =2,4t,5,8,77,377  (n<=1000)
NWpm(n)=(2 n + n)*2n - 1 prime for n =1b,2,34,107,1568,1933,3551,6793 (n<=10000 with help of Gary Chaffey)
NCmp(n)=(2 n + n)*2n + 1 prime for n =1b,3,6,14,21,27,51,61,103,123,126,414,499 (n<=1000)


New "Near-Cullen" and "Near-Woodall" :

(2 n - n+1)*2n + 1 prime for n =1t,2t,39,44,62 (n<=1000)
(2 n - n-1)*2n + 1 prime for n = 2t,7t,11,13,14,20,37,53,71,132,140,613,641,665,757,788 (n<=1000)
(2 n - n+1)*2n - 1 prime for n =1t,2t,3,8,14,35,75,83,89,90,215,342,753 (n<=1000)
(2 n - n-1)*2n - 1 prime for n = 2t,3,7t (n<=1000)
(2 n + n+1)*2n - 1 prime for n =1,13 (n<=1000)
(2 n + n-1)*2n - 1 prime for n = 1t,2,3,5t,9,18,30,48,54,278,450,464 (n<=1000)
(2 n + n+1)*2n + 1 prime for n = 2,3,4,5,15,23,53,57,75,233,464,671 (n<=1000)
(2 n + n-1)*2n + 1 prime for pour n = 1t,5t,49,269 (n<=1000)


Top :

Form
Digits Who When
Nmp(k,n) (2^19592-16)*(2^19592)+1 11796 Henri Lifchitz 12/08/1998
Nmm(k,n) (2^9586-34)*(2^9586)-1 5772 Didier Boivin 17/02/2002
Npm(k,n) (2^978+8)*(2^978)-1 589 Henri Lifchitz 12/08/1998
Npp(k,n) (2^2604+6)*(2^2604)+1 1568 Henri Lifchitz 12/08/1998
Twin (2^309-71)*(2^309)-1
(2^309-71)*(2^309)+1
187 (p) Gary Chaffey 30/06/2005
Sophie Germain (2^130-7228)*(2^130)-1
2*((2^130-7228)*(2^130)-1)+1
79 (p) Gary Chaffey 30/06/2005
BiTwin (2^10-169)*(2^10)+/- 1
2*(2^10-169)*(2^10)+/- 1
6 (p) Gary Chaffey 30/06/2005
Nouveau Woodall (2^6793+6793)*(2^6793)-1 4090 Gary Chaffey  
Nouveau Cullen (2^21143-21143)*(2^21143)+1 12730 Henri Lifchitz 10/05/2004
NCullen&NWoodall = Twin (2^4-4)*(2^4)+1
(2^4-4)*(2^4)-1
3 Henri Lifchitz 12/08/1998

CC length 2
(type 2)

(2^747-4)*(2^747) + 1
2*((2^747 - 4)*(2^747 + 1) -1
450 (p) Henri Lifchitz 12/08/1998

CC length 3
(type 1)

(2^31-1718)*(2^31)-1
2*((2^31-1718)*(2^31)-1)+1
4*((2^31-1718)*(2^31)-1)+3
19 (p) Gary Chaffey 30/06/2005
CC length 4
(type 1)

(2^20-8161)*(2^20)-1
2*((2^20-8161)*(2^20)-1)+1
4*((2^20-8161)*(2^20)-1)+3
8*((2^20-8161)*(2^20)-1)+7

13 (p) Gary Chaffey 30/06/2005
Triplets

(2^15-31847)*(2^15)-1
(2^15-31847)*(2^15)+1
(2^15-31847)*(2^15)+5

8 (p) Gary Chaffey 30/06/2005

 


Generalization:   (bn+-k)*bn+-1

Those numbers have similar properties with (2^n+-k)*2^n+-1, for their forms of prime factors, and in particular if k=1 and n=p^m, we recognize some properties of the Generalized Fermat numbers. A more accurate study will be made as soon as possible.

Didier Boivin has checked the primality of these numbers for b from 3 to 7, n from 1 to 1000, and k from 1 to 20. (Word document).


You can also consult the next link :


Created by  Henri Lifchitz : October, 12 1998,  last modification: May, 22 2011.