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Old 2017-08-12, 03:09   #34
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Searched up to b=260.
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Old 2017-08-12, 15:23   #35
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Searched up to b=300.
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Old 2017-08-15, 16:34   #36
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Searched up to b=432.

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Old 2017-09-21, 16:21   #37
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Links for the full list for the n's <= 1000 such that (b+-1)*b^n+-1 is prime for bases b <= 1000:

(b-1)*b^n-1 case: http://www.bitman.name/math/table/484
(b-1)*b^n+1 case: http://www.bitman.name/math/table/477
(b+1)*b^n-1 case: http://www.bitman.name/math/table/471
(b+1)*b^n+1 case: http://www.bitman.name/math/table/474

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Old 2017-10-17, 22:41   #38
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Searched the dual form (i.e. b^n+-b+-1) for bases 2<=b<=48.

Note 1: The name of the file is not right, it should be "numbers n such that b^n+-b+-1 is (probable) prime for b up to 48".

Note 2: In this file, "b, +-1, +-1" means "b^n+-b+-1", e.g. "12, +1, -1" means "12^n+12-1", "19, -1, +1" means "19^n-19+1".

Note 3: If b = 1 (mod 3), then all numbers of the form b^n+b+1 are divisible by 3, thus "b, +1, +1" should not be searched for b = 1 (mod 3).

Note 4: Some large terms may be only probable primes (i.e. not definite primes), since for these numbers N, neither N-1 nor N+1 can be trivially written into a product.
Attached Files
File Type: txt least k such that n^k+-n+-1 is prime.txt (8.4 KB, 52 views)

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Old 2017-10-17, 23:08   #39
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Quote:
Originally Posted by sweety439 View Post
Searched the dual form (i.e. b^n+-b+-1) for bases 2<=b<=48.

Note 1: The name of the file is not right, it should be "numbers n such that b^n+-b+-1 is (probable) prime for b up to 48".

Note 2: In this file, "b, +-1, +-1" means "b^n+-b+-1", e.g. "12, +1, -1" means "12^n+12-1", "19, -1, +1" means "19^n-19+1".

Note 3: If b = 1 (mod 3), then all numbers of the form b^n+b+1 are divisible by 3, thus "b, +1, +1" should not be searched for b = 1 (mod 3).

Note 4: Some large terms may be only probable primes (i.e. not definite primes), since for these numbers N, neither N-1 nor N+1 can be trivially written into a product.
The dual of (b-1)*b^n-1 (b-- for original form) is b^n-(b-1) = b^n-b+1 (b-+ for dual form)
The dual of (b-1)*b^n+1 (b-+ for original form) is b^n+(b-1) = b^n+b-1 (b+- for dual form)
The dual of (b+1)*b^n-1 (b+- for original form) is b^n-(b+1) = b^n-b-1 (b-- for dual form)
The dual of (b+1)*b^n+1 (b++ for original form) is b^n+(b+1) = b^n+b+1 (b++ for dual form)

A form and its dual form have the same Nash weight, thus, the possibility for them to have a prime in the given interval of n are the same.

Last fiddled with by sweety439 on 2017-10-19 at 04:18
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Old 2017-10-18, 01:31   #40
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Dual form searched up to b=160, up to n=1024.
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File Type: txt numbers k such that n^k+-n+-1 is prime.txt (23.0 KB, 54 views)

Last fiddled with by sweety439 on 2017-10-18 at 02:53
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Old 2017-10-18, 01:59   #41
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For dual form, please see these sequences:

b^n-b-1:

b=2: http://oeis.org/A050414
b=3: http://oeis.org/A058959
b=4: http://oeis.org/A217348
b=5: http://oeis.org/A165701
b=6: http://oeis.org/A217352
b=7: http://oeis.org/A217131
b=8: http://oeis.org/A217383
b=9: http://oeis.org/A217493
b=10: http://oeis.org/A092767
b=11: (sequence not in OEIS)
b=12: (sequence not in OEIS)

b^n-b+1:

b=2: http://oeis.org/A000043
b=3: http://oeis.org/A014224
b=4: http://oeis.org/A059266
b=5: http://oeis.org/A059613
b=6: http://oeis.org/A059614
b=7: http://oeis.org/A191469
b=8: http://oeis.org/A217380
b=9: http://oeis.org/A177093
b=10: http://oeis.org/A095714
b=11: (sequence not in OEIS)
b=12: (sequence not in OEIS)

b^n+b-1:

b=2: (sequence not in OEIS)
b=3: http://oeis.org/A051783
b=4: http://oeis.org/A089437
b=5: http://oeis.org/A124621
b=6: http://oeis.org/A145106
b=7: http://oeis.org/A217130
b=8: http://oeis.org/A217381
b=9: http://oeis.org/A217385
b=10: http://oeis.org/A088275
b=11: (sequence not in OEIS)
b=12: http://oeis.org/A137654

b^n+b+1:

b=2: http://oeis.org/A057732
b=3: http://oeis.org/A058958
b=5: http://oeis.org/A089142
b=6: http://oeis.org/A217351
b=8: http://oeis.org/A217382
b=9: http://oeis.org/A217492
b=11: (sequence not in OEIS)
b=12: (sequence not in OEIS)

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Old 2017-10-18, 02:09   #42
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Also these sequences:

Least k>=1 such that n^k-n-1 is prime: http://oeis.org/A178250

Least k>=1 such that n^k-n+1 is prime: http://oeis.org/A113516

Least k>=1 such that n^k+n-1 is prime: http://oeis.org/A076845

Least k>=1 such that n^k+n+1 is prime: (sequence not in OEIS)

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Old 2017-10-18, 02:35   #43
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Quote:
Originally Posted by sweety439 View Post
For the primes of the form (b+-1)*b^n+-1 with integer b>=2 and integer n>=1:

Nmm(b,n) = (b-1)*b^n-1 is already searched in http://harvey563.tripod.com/wills.txt, but one prime is missing in this website: (91-1)*91^519-1, and the exponent of b=38 is wrong, it should be (38-1)*38^136211-1, not (38-1)*38^136221-1. Besides, (128-1)*128^n-1 has been reserved by Cruelty. The known primes with b<=500 and n>1000 are (38-1)*38^136211-1, (83-1)*83^21495-1, (98-1)*98^4983-1, (113-1)*113^286643-1, (125-1)*125^8739-1, (188-1)*188^13507-1, (228-1)*228^3695-1, (347-1)*347^4461-1, (357-1)*357^1319-1, (401-1)*401^103669-1, (417-1)*417^21002-1, (443-1)*443^1691-1, (458-1)*458^46899-1, (494-1)*494^21579-1. The bases b<=500 without known prime are 128 (n>1700000), 233, 268, 293, 383, 478, 488, all are checked to at least n=200000.

Nmp(b,n) = (b-1)*b^n+1, the known primes with b<=500 and n>500 are (53-1)*53^960+1, (65-1)*65^946+1, (77-1)*77^828+1, (88-1)*88^3022+1, (122-1)*122^6216+1, (158-1)*158^1620+1, (180-1)*180^2484+1, (197-1)*197^520+1, (248-1)*248^604+1, (249-1)*249^1851+1, (257-1)*257^1344+1, (269-1)*269^1436+1, (275-1)*275^980+1, (319-1)*319^564+1, (356-1)*356^528+1, (434-1)*434^882+1. The bases b<=500 without known prime are 123 (n>100000), 202 (reserving, n>1024), 251 (n>73000), 272 (reserving, n>1024), 297 (CRUS prime), 298, 326, 328, 342 (n>100000), 347, 362, 363, 419, 422, 438 (n>100000), 452, 455, 479, 487 (n>100000), 497, 498 (CRUS prime), all are checked to at least n=1024.

Npm(b,n) = (b+1)*b^n-1, the known primes with b<=300 and n>500 are (63+1)*63^1483-1, (88+1)*88^1704-1, (143+1)*143^921-1. The bases b<=300 without known primes are 208, 232, 282, 292, all are checked to at least n=1024. (except the case b=208, all of them are CRUS primes)

Npp(b,n) = (b+1)*b^n+1, in this case this b should not = 1 (mod 3), or all numbers of the form (b+1)*b^n+1 are divisible by 3, the known primes with b<=200 (b != 1 mod 3) and n>500 are (171+1)*171^1851+1, there is no such prime with b=201 and n<=1024.
There are also OEIS sequences for the smallest n such that (b-1)*b^n+-1 is prime for prime b:

(b-1)*b^n-1 case: http://oeis.org/A122396 (the "n" in this formula is one less than the number in the sequence)
(b-1)*b^n+1 case: http://oeis.org/A087139 (the "n" in this formula is one less than the number in the sequence)
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Old 2017-10-18, 02:42   #44
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Quote:
Originally Posted by sweety439 View Post
The dual of (b-1)*b^n-1 (b-- for original form) is b^n-(b-1) = b^n-b+1 (b-+ for dual form)
The dual of (b-1)*b^n+1 (b-+ for original form) is b^n+(b-1) = b^n+b-1 (b+- for dual form)
The dual of (b+1)*b^n-1 (b+- for original form) is b^n-(b+1) = b^n-b-1 (b-- for dual form)
The dual of (b+1)*b^n+1 (b++ for original form) is b^n+(b+1) = b^n+b+1 (b++ for dual form)
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
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