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Old 2017-10-18, 02:47   #45
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e.g.

The dual of 11*12^n-1 is 12^n-11
The dual of 11*12^n+1 is 12^n+11
The dual of 13*12^n-1 is 12^n-13
The dual of 13*12^n+1 is 12^n+13
In original file:

"12, -1, -1" means 11*12^n-1
"12, -1, +1" means 11*12^n+1
"12, +1, -1" means 13*12^n-1
"12, +1, +1" means 13*12^n+1

In dual file:

"12, -1, -1" means 12^n-13
"12, -1, +1" means 12^n-11
"12, +1, -1" means 12^n+11
"12, +1, +1" means 12^n+13
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Old 2017-10-18, 02:54   #46
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Dual form searched up to b=160, up to n=1024.
There are (probable) primes found by the OEIS sequences with n>1024:

71^3019-71+1
88^2848-88+1

See the OEIS sequence, 93^n-93+1 has been searched to n=60K with no (probable) prime found.

Also, recently, I found the (probable) prime 107^1400+107-1

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Old 2017-10-19, 04:13   #47
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Now, I name these numbers:

(b-1)*b^n-1: Williams primes of 1st kind base b
(b-1)*b^n+1: Williams primes of 2nd kind base b
(b+1)*b^n-1: 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)

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Old 2017-10-19, 04:14   #48
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Now, I name these numbers:

(b-1)*b^n-1: Williams primes of 1st kind base b
(b-1)*b^n+1: Williams primes of 2nd kind base b
(b+1)*b^n-1: Williams primes of 3rd kind base b
(b+1)*b^n+1: Williams primes of 4th kind base b
Thus, for the dual of these numbers:

b^n-(b-1): dual Williams primes of 1st kind base b
b^n+(b-1): 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)

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Old 2017-10-19, 04:15   #49
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Now, I name these numbers:

(b-1)*b^n-1: Williams primes of 1st kind base b
(b-1)*b^n+1: Williams primes of 2nd kind base b
(b+1)*b^n-1: Williams primes of 3rd kind base b
(b+1)*b^n+1: Williams primes of 4th kind base b
For the file in post #36:

"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

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Old 2017-10-19, 04:20   #50
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Thus, for the dual of these numbers:

b^n-(b-1): dual Williams primes of 1st kind base b
b^n+(b-1): 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
For the file in post #40:

"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

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Old 2017-12-03, 07:14   #51
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Update the file for the 1 <= n <= 1024 such that these numbers are primes for bases 2<=b<=512.

1st: (b-1)*b^n-1
2nd: (b-1)*b^n+1
3rd: (b+1)*b^n-1
4th: (b+1)*b^n+1 (only for the bases b not = 1 mod 3)
1st dual: b^n-(b-1)
2nd dual: b^n+(b-1)
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^n-1 is b^n-k.
Attached Files
File Type: txt list of Williams primes.txt (120.4 KB, 166 views)

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Old 2017-12-03, 07:36   #52
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Originally Posted by sweety439 View Post
Update the file for the 1 <= n <= 1024 such that these numbers are primes for bases 2<=b<=512.

1st: (b-1)*b^n-1
2nd: (b-1)*b^n+1
3rd: (b+1)*b^n-1
4th: (b+1)*b^n+1 (only for the bases b not = 1 mod 3)
1st dual: b^n-(b-1)
2nd dual: b^n+(b-1)
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^n-1 is b^n-k.
Blank rows for b<=200 (for 1<=n<=1024):

38, 1st (37*38^n-1): The first prime is at n=136211.
63, 3rd (64*63^n-1): The first prime is at n=1483.
71, 1st dual (71^n-70): The first (probable) prime is at n=3019.
83, 1st (82*83^n-1): The first prime is at n=21495.
88, 2nd (87*88^n+1): The first prime is at n=3022.
88, 3rd (89*88^n-1): The first prime is at n=1704.
88, 1st dual (88^n-87): The first (probable) prime is at n=2848.
93, 1st dual (93^n-92): No (probable) prime found for n<=60000.
98, 1st (97*98^n-1): 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^n-1): The first prime is at n=286643.
113, 1st dual (113^n-112): 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^n-1): The first prime is at n=8739.
128, 1st (127*128^n-1): No prime found for n<=1700000.
152, 1st dual (152^n-151): 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^n-157): 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^n-1): The first prime is at n=13507.
188, 1st dual (188^n-187): Only searched up to n=1024, no further searching.
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Old 2017-12-03, 07:53   #53
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Blank rows for b<=200 (for 1<=n<=1024):

38, 1st (37*38^n-1): The first prime is at n=136211.
63, 3rd (64*63^n-1): The first prime is at n=1483.
71, 1st dual (71^n-70): The first (probable) prime is at n=3019.
83, 1st (82*83^n-1): The first prime is at n=21495.
88, 2nd (87*88^n+1): The first prime is at n=3022.
88, 3rd (89*88^n-1): The first prime is at n=1704.
88, 1st dual (88^n-87): The first (probable) prime is at n=2848.
93, 1st dual (93^n-92): No (probable) prime found for n<=60000.
98, 1st (97*98^n-1): 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^n-1): The first prime is at n=286643.
113, 1st dual (113^n-112): 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^n-1): The first prime is at n=8739.
128, 1st (127*128^n-1): No prime found for n<=1700000.
152, 1st dual (152^n-151): 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^n-157): 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^n-1): The first prime is at n=13507.
188, 1st dual (188^n-187): Only searched up to n=1024, no further searching.
It is highly conjectured that all bases b>=2 have infinitely many (dual) Williams primes of the 1st/2nd/3rd kind, and all bases b>=2 not = 1 mod 3 have infinitely many (dual) Williams primes of the 4th kind. For every given kind for (dual) Williams prime, only few bases b>=2 have no known (probable) prime.

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 N-1 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 N-1 nor N+1 can be trivially written into a product.

Last fiddled with by sweety439 on 2017-12-09 at 14:43
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Old 2017-12-03, 07:57   #54
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Update the file for the 1 <= n <= 1024 such that these numbers are primes for bases 2<=b<=512.

1st: (b-1)*b^n-1
2nd: (b-1)*b^n+1
3rd: (b+1)*b^n-1
4th: (b+1)*b^n+1 (only for the bases b not = 1 mod 3)
1st dual: b^n-(b-1)
2nd dual: b^n+(b-1)
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^n-1 is b^n-k.
Reserve 513<=b<=1024, also for 1 <= n <= 1024.
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Old 2017-12-03, 23:34   #55
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Blank rows for b<=200 (for 1<=n<=1024):

38, 1st (37*38^n-1): The first prime is at n=136211.
63, 3rd (64*63^n-1): The first prime is at n=1483.
71, 1st dual (71^n-70): The first (probable) prime is at n=3019.
83, 1st (82*83^n-1): The first prime is at n=21495.
88, 2nd (87*88^n+1): The first prime is at n=3022.
88, 3rd (89*88^n-1): The first prime is at n=1704.
88, 1st dual (88^n-87): The first (probable) prime is at n=2848.
93, 1st dual (93^n-92): No (probable) prime found for n<=60000.
98, 1st (97*98^n-1): 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^n-1): The first prime is at n=286643.
113, 1st dual (113^n-112): 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^n-1): The first prime is at n=8739.
128, 1st (127*128^n-1): No prime found for n<=1700000.
152, 1st dual (152^n-151): 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^n-157): 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^n-1): The first prime is at n=13507.
188, 1st dual (188^n-187): Only searched up to n=1024, no further searching.
Dual n's for these forms:

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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