New forms of primes

Let the numbers of the following forms:
• Nmp(k,n)  =  (2 n - k)*2n + 1
• Nmm(k,n) =  (2 n - k)*2n -  1
• Npp(k,n)  =  (2 n +k)*2n  + 1
• Npm(k,n)  =  (2 n + k)*2n - 1

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:
• It exists Twins (Nmp(k,n) and Nmm(k,n) prime or Npp(k,n) and Npm(k,n) prime)
• It exists Sierpinski, Riesel and Brier k. Morever, it is unnecessary to find a long time the smallest Sierpinski (k=5 +15a for Nmp form and k=10+15a for Npp form) and the smallest Brier (3+3a for Nmm and Npm forms).
For the Nmp  form k=7,13,34,79,. and for Npp form k=15,.. perhaps are also Sierpinski ?
• It exists Cullen (k=n in all the four forms).
• It exists Sophie Germain and Cunningham chains, but they are "visually" more difficult to track down.
• Nmp(1,n) can only be prime if n=2a3b with a>=1 and b>=0. In practice it is only prime for n=1,2,4,32 (n<=400000). The divisors of these numbers are of the form 6nc+1 !!, which can allow the quick search of the composite numbers.  The resemblance with Fermat numbers behaviour (22^n+1) is "strange" and the primality for a n (with the very restrictive form 2a3b) greater than 400000 would entail probably a record.
• Similarly Npp(1,n) is prime only for n=1,3,9 (n<=40000) !

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.