20171018, 02:47  #45  
Nov 2016
8B7_{16} Posts 
Quote:
"12, 1, 1" means 11*12^n1 "12, 1, +1" means 11*12^n+1 "12, +1, 1" means 13*12^n1 "12, +1, +1" means 13*12^n+1 In dual file: "12, 1, 1" means 12^n13 "12, 1, +1" means 12^n11 "12, +1, 1" means 12^n+11 "12, +1, +1" means 12^n+13 

20171018, 02:54  #46 
Nov 2016
23·97 Posts 
There are (probable) primes found by the OEIS sequences with n>1024:
71^301971+1 88^284888+1 See the OEIS sequence, 93^n93+1 has been searched to n=60K with no (probable) prime found. Also, recently, I found the (probable) prime 107^1400+1071 Last fiddled with by sweety439 on 20171018 at 02:56 
20171019, 04:13  #47 
Nov 2016
23×97 Posts 
Now, I name these numbers:
(b1)*b^n1: Williams primes of 1st kind base b (b1)*b^n+1: Williams primes of 2nd kind base b (b+1)*b^n1: Williams primes of 3rd kind base b (b+1)*b^n+1: Williams primes of 4th kind base b (the Williams primes of 4th kind base b exist only if b not = 1 mod 3) Last fiddled with by sweety439 on 20171019 at 04:24 
20171019, 04:14  #48  
Nov 2016
4267_{8} Posts 
Quote:
b^n(b1): dual Williams primes of 1st kind base b b^n+(b1): dual Williams primes of 2nd kind base b b^n(b+1): dual Williams primes of 3rd kind base b b^n+(b+1): dual Williams primes of 4th kind base b (similarly, the dual Williams primes of 4th kind base b exist only if b not = 1 mod 3) Last fiddled with by sweety439 on 20171019 at 04:24 

20171019, 04:15  #49  
Nov 2016
23×97 Posts 
Quote:
"b, 1, 1" is Williams primes of 1st kind base b "b, 1, +1" is Williams primes of 2nd kind base b "b, +1, 1" is Williams primes of 3rd kind base b "b, +1, +1" is Williams primes of 4th kind base b Last fiddled with by sweety439 on 20171019 at 04:20 

20171019, 04:20  #50  
Nov 2016
23×97 Posts 
Quote:
"b, 1, 1" is dual Williams primes of 3rd kind base b "b, 1, +1" is dual Williams primes of 1st kind base b "b, +1, 1" is dual Williams primes of 2nd kind base b "b, +1, +1" is dual Williams primes of 4th kind base b Last fiddled with by sweety439 on 20171019 at 04:21 

20171203, 07:14  #51 
Nov 2016
23·97 Posts 
Update the file for the 1 <= n <= 1024 such that these numbers are primes for bases 2<=b<=512.
1st: (b1)*b^n1 2nd: (b1)*b^n+1 3rd: (b+1)*b^n1 4th: (b+1)*b^n+1 (only for the bases b not = 1 mod 3) 1st dual: b^n(b1) 2nd dual: b^n+(b1) 3rd dual: b^n(b+1) 4th dual: b^n+(b+1) (only for the bases b not = 1 mod 3) Note that a form (1st/2nd/3rd/4th) and its dual have the same Nash weight, since the dual for the form k*b^n+1 is b^n+k, and the dual for the form k*b^n1 is b^nk. Last fiddled with by sweety439 on 20171203 at 07:15 
20171203, 07:36  #52  
Nov 2016
23×97 Posts 
Quote:
38, 1st (37*38^n1): The first prime is at n=136211. 63, 3rd (64*63^n1): The first prime is at n=1483. 71, 1st dual (71^n70): The first (probable) prime is at n=3019. 83, 1st (82*83^n1): The first prime is at n=21495. 88, 2nd (87*88^n+1): The first prime is at n=3022. 88, 3rd (89*88^n1): The first prime is at n=1704. 88, 1st dual (88^n87): The first (probable) prime is at n=2848. 93, 1st dual (93^n92): No (probable) prime found for n<=60000. 98, 1st (97*98^n1): The first prime is at n=4983. 107, 2nd dual (107^n+106): The first (probable) prime is at n=1400. 113, 1st (112*113^n1): The first prime is at n=286643. 113, 1st dual (113^n112): Only searched up to n=1024, no further searching. 113, 2nd dual (113^n+112): Only searched up to n=1024, no further searching. 122, 2nd (121*122^n+1): The first prime is at n=6216. 123, 2nd (122*123^n+1): No prime found for n<=100000. 123, 2nd dual (123^n+122): Only searched up to n=1024, no further searching. 125, 1st (124*125^n1): The first prime is at n=8739. 128, 1st (127*128^n1): No prime found for n<=1700000. 152, 1st dual (152^n151): Only searched up to n=1024, no further searching. 158, 2nd (157*158^n+1): The first prime is at n=1620. 158, 1st dual (158^n157): Only searched up to n=1024, no further searching. 171, 4th (172*171^n+1): The first prime is at n=1851. 173, 2nd dual (173^n+172): Only searched up to n=1024, no further searching. 179, 2nd dual (179^n+178): Only searched up to n=1024, no further searching. 180, 2nd (179*180^n+1): The first prime is at n=2484. 188, 1st (187*188^n1): The first prime is at n=13507. 188, 1st dual (188^n187): Only searched up to n=1024, no further searching. 

20171203, 07:53  #53  
Nov 2016
4267_{8} Posts 
Quote:
All bases 2<=b<=122 have at least one known Williams (probable) prime either original or dual for every given kind, this is the "mixed Williams (probable) prime problem", the first base b>=2 with neither known Williams prime nor known dual Williams (probable) prime for some given kind is b=123 for the 2nd kind. Note: For every kind, the original Williams prime have no pseudoprimes, since either N1 or N+1 can be trivially written into a product. However, for every kind, when n is large, the dual Williams prime cannot be proven to be prime easily (except the case b=2 for the 1st kind and the 2nd kind, which dual Williams primes for a given kind are the same as the original Williams primes for the same kind), since neither N1 nor N+1 can be trivially written into a product. Last fiddled with by sweety439 on 20171209 at 14:43 

20171203, 07:57  #54  
Nov 2016
100010110111_{2} Posts 
Quote:


20171203, 23:34  #55  
Nov 2016
23·97 Posts 
Quote:
38, 1st dual: 5, 429 63, 3rd dual: 7, 19 71, 1st: 1, 59, 93 83, 1st dual: 965 88, 2nd dual: 8, 20, 974 88, 3rd dual: 11, 31, 36, 809, 831 88, 1st: 3, 163, 366, 522 93, 1st: 476, 908 98, 1st dual: 5, 201, 445, 449 107, 2nd: 4 (113, 1st both sides have no prime for 1<=n<=1024) 113, 2nd: 4, 16, 26, 236 122, 2nd dual: 4, 8, 60, 568 (123, 2nd both sides have no prime for 1<=n<=1024) 125, 1st dual: 3, 25, 287 128, 1st dual: 401 152, 1st: 3, 15, 55, 143, 355 158, 2nd dual: 2 158, 1st: 127, 263, 323 171, 4th dual: 4, 15, 42, 132, 364, 471 173, 2nd: 2, 70, 114 179, 2nd: 46, 550, 832 180, 2nd dual: 1, 2, 7, 8 (188, 1st both sides have no prime for 1<=n<=1024) 

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