Primes a^8*2^8m+1 = (a*2^m)^8+1 for odd prime powers a < 72, m < 42,500:
(9*2^m)^8+1 : m=5,22,224,281,2492,3239,10940,13833,21060,45120*
(17*2^m)^8+1 : m=8,12,15,25,105,191,260,393,466,682,743,990,8219,13450
(19*2^m)^8+1 : m=3,54,736,2574,10440,15597
(25*2^m)^8+1 : m=5,7,36,158,912,1893,2456,7377,20061,47004
(43*2^m)^8+1 : m=19,22,28,36,37,43,99,175,303,427,536,592,4352,6268,13021,24615
(47*2^m)^8+1 : m=8,13,16,94,1750,5035,6457,13273,19855,28284
(49*2^m)^8+1 : m=3,13,17,35,93,365,675,1031,2025,2207,2991,3657,17295
(53*2^m)^8+1 : m=10,45,71,124,244,255,429,469,2004,2979,14914,37463
(59*2^m)^8+1 : m=1,10,52,87,211,345,762,2337,18244
(67*2^m)^8+1 : m=32,174,292,344,1957,2919,15900,19816,21560,31672,38633
In standard base 2 format (n=8m, n < 340,000):
9^8*2^n+1 : n=40,176,1792,2248,19936,25912,87520,110664,168480,360960*
17^8*2^n+1 : n=64,96,120,200,840,1528,2080,3144,3728,5456,5944,7920,65752,
107600
19^8*2^n+1 : n=24,432,5888,20592,83520,124776
25^8*2^n+1 : n=40,56,288,1264,7296,15144,19648,59016,160488,376032
43^8*2^n+1 : n=152,176,224,288,296,344,792,1400,2424,3416,4288,4736,34816,
50144,104168,196920
47^8*2^n+1 : n=64,104,128,752,14000,40280,51656,106184,158840,226272
49^8*2^n+1 : n=24,104,136,280,744,2920,5400,8248,16200,17656,23928,29256,
138360
53^8*2^n+1 : n=80,360,568,992,1952,2040,3432,3752,16032,23832,119312,299704
59^8*2^n+1 : n=8,80,416,696,1688,2760,6096,18696,145952
67^8*2^n+1 : n=256,1392,2336,2752,15656,23352,127200,158528,172480,253376,
309064
*This prime is from the subsequence (3*2^m)^16+1 = 3^16*2^n+1 which has been
tested to higher levels. (n > 1,000,000)
The siever needs a^8 written out in full:
9^8 = 43046721
17^8 = 6975757441
19^8 = 16983563041
25^8 = 152587890625
43^8 = 11688200277601
47^8 = 23811286661761
49^8 = 33232930569601
53^8 = 62259690411361
59^8 = 146830437604321
67^8 = 406067677556641
Recent primes:
8 Jan 2007 62259690411361*2^299704+1 is prime!
11 Feb 2007 406067677556641*2^309064+1 is prime!
24 Mar 2007 152587890625*2^376032+1 is prime!
The choice of a < 72, exponent=8 allows efficient sieving (all factors are
of the form 16k+1), but a^8 is not so large that LLR has to use a slow
generic FFT. (Using exponent=16 would reduce the choices for a to 3 or 5).
n=1,000,000 takes about 10,000 P4 GHz seconds to test with LLR 3.7.1.
Sieving speed for all 10 sequences 200,000 < n < 1,000,000 is about 1.35
million p per P3 GHz second. (At p=2.5 trillion with srsieve 0.5.2).
(
geocities.com/g_w_reynolds)