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Old 2016-12-08, 19:10   #1
sweety439
 
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Default Primes of the form (b+-1)*b^n+-1 and b^n+-(b+-1)

For (b-1)*b^n-1, called Williams prime base b, are already searched in http://harvey563.tripod.com/wills.txt for b<=2049, but no prime was found for b = 128, 233, 268, 293, 383, 478, 488, ...

Are there also a project to search primes of the form (b-1)*b^n+1? I searched it for bases b up to 500, but found no primes for b = 122, 123, 180, 202, 249, 251, 257, 269, 272, 297, 298, 326, 328, 342, 347, 362, 363, 419, 422, 438, 452, 455, 479, 487, 497, 498.
Attached Files
File Type: txt least k such that (n-1)n^k+1 is prime.txt (3.6 KB, 36 views)

Last fiddled with by sweety439 on 2017-12-20 at 22:25
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Old 2016-12-08, 19:17   #2
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OEIS sequence for n's such that (b-1)*b^n+-1 is prime:

(b-1)*b^n-1 case:

b=2: A000043

b=3: A003307

b=4: A272057

b=5: A046865

b=6: A079906

b=7: A046866

b=8: A268061

b=9: A268356

b=10: A056725

b=11: A046867

b=12: A079907

(b-1)*b^n+1 case:

b=2: sequence is not in OEIS.

b=3:  A003306

b=4: sequence is not in OEIS.

b=5: A204322

b=6: A247260

b=7: A245241

b=8: A269544

b=9: A056799

b=10: A056797

b=11: A057462

b=12: A251259
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Old 2016-12-08, 19:36   #3
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Quote:
Originally Posted by sweety439 View Post
b=2: sequence is not in OEIS.
A019434
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Old 2016-12-08, 19:54   #4
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Quote:
Originally Posted by sweety439 View Post
b=4: sequence is not in OEIS.
subset of A039687 if they exist at all.

doh misread what you meant. as did paul I believe.

https://oeis.org/A002253 comes up through a few clicks and searching the cf and adjacent sequences though.

Last fiddled with by science_man_88 on 2016-12-08 at 20:12
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Old 2016-12-14, 19:42   #5
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See http://mersenneforum.org/showthread....=286643&page=5.

These are text files for the smallest n such that (b+1)*b^n-1 or (b+1)*b^n+1 is prime. (For the case (b+1)*b^n+1, b should not = 1 (mod 3), or all the numbers of this form are divisible by 3 and cannot be prime)
Attached Files
File Type: txt least k such that (n+1)n^k-1 is prime.txt (2.0 KB, 43 views)
File Type: txt least k such that (n+1)n^k+1 is prime.txt (882 Bytes, 45 views)
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Old 2016-12-15, 14:18   #6
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For (b-1)*b^n-1, in the website http://harvey563.tripod.com/wills.txt for b<=2049, a prime is missing: (91-1)*91^519-1.

Also the primes with exponent > 1000: (up to b=500)

(38-1)*38^136211-1 (this website wrongly writes the exponent as 136221)
(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

Last fiddled with by sweety439 on 2016-12-15 at 14:20
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Old 2016-12-15, 14:24   #7
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OEIS sequence only has few sequences for n's such that (b+1)*b^n+-1 is prime:

(b+1)*b^n-1 case:

b=2: A002235
b=3: A005540
b=5: A257790
b=10: A111391

(b+1)*b^n+1 case: (in this case, b should not = 1 (mod 3), or all the numbers of this form are divisible by 3 and cannot be prime)

b=2: A002253
b=3: A005537
b=5: A143279

Last fiddled with by sweety439 on 2016-12-15 at 14:27
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Old 2017-01-02, 17:50   #8
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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.

Last fiddled with by sweety439 on 2017-01-02 at 18:25
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Old 2017-01-04, 11:21   #9
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hi,
i do search for williams primes
2017*2018^n-1
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Old 2017-01-04, 16:30   #10
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Quote:
Originally Posted by lalera View Post
hi,
i do search for williams primes
2017*2018^n-1
No, base 2017 is (2017-1)*2017^n-1, you searched base 2018.

You can also search (2017-1)*2017^n+1 and (2017+1)*2017^n-1. (please do not search (2017+1)*2017^n+1, since all such numbers of this form are divisible by 3, there is no prime of this form)
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Old 2017-01-04, 17:15   #11
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hi,
i gave the formula i do search for
2017*2018^n-1

Last fiddled with by lalera on 2017-01-04 at 17:17
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