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2020-07-21, 03:28   #23
sweety439

Nov 2016

22·3·197 Posts

Quote:
 Originally Posted by kar_bon There're two problems with the example of 13*43^n-1: 1. All candidates are divisible by 6. 2. The smallest p-value to start with srsieve is p=44, so have to be greater than the base.
No, it just have to > all prime factors of gcd(k+-1,b-1), thus we can sieve start with p=5 (the gcd is 6, thus we should not sieve the primes 2 and 3), like the case of R36 (which can use the current srsieve, since 36 is even, the current srsieve can be used if and only if at least one of k and b is even), we can sieve R36 start with p=11, since we should not sieve the primes 5 and 7, it need not to be > 36

2020-07-21, 03:40   #24
gd_barnes

May 2007
Kansas; USA

3×7×487 Posts

Quote:
 Originally Posted by sweety439 The divisor of k*b^n+-1 (+ for Sierp, - for Riesel) is always gcd(k+-1,b-1) (+ for Sierp, - for Riesel) (see post https://mersenneforum.org/showpost.p...&postcount=230), since gcd(k+-1,b-1) is the trivial factor of k*b^n+-1, it is simply to take out this factor, thus, the divisor of R43, k=13 is 6, not 2 (gcd(13-1,43-1) = 6), and the formula of R43, k=13 is (13*43^n-1)/6, not (13*43^n-1)/2
Of course you are correct. I was only giving a simplified example. Even though the divisor here is 6 we still have the same problem with srsieve or srsieve2 giving an immediate error when all candidates are divisible by 2.

We simply need this error check removed from srsieve and/or srsieve2.

Last fiddled with by gd_barnes on 2020-07-21 at 03:41

2020-07-21, 03:49   #25
sweety439

Nov 2016

1001001111002 Posts

Quote:
 Originally Posted by sweety439 Well, you said these conjectures would have to have multiple and separate sieves done (see your post https://mersenneforum.org/showpost.p...&postcount=243), however, you can use srsieve to sieve the sequence k*b^n+-1 for primes p not dividing gcd(k+-1,b-1), and initialized the list of candidates to not include n for which there is some prime p dividing gcd(k+-1,b-1) for which p dividing (k*b^n+-1)/gcd(k+-1,b-1) (like we can initialized the list of candidates to not include n for which k*b^n+-1 has algebra factors, e.g. for square k's for k*b^n-1, we can remove all even n in the sieve file, and for cube k's for k*b^n-1 and k*b^n+1, we can remove all n divisible by 3 in the sieve file)
In fact, for Sierpinski/Riesel base b, we can use srsieve to sieve the sequence k*b^n+-1 for primes p not dividing b-1 (since gcd(k*b^n+-1,b-1) = gcd(k+-1,b-1) for all n), and initialized the list of candidates to not include n for which there is some prime p dividing b-1 for which p dividing (k*b^n+-1)/gcd(k+-1,b-1)

2020-07-21, 07:41   #26
kar_bon

Mar 2006
Germany

54428 Posts

Quote:
 Originally Posted by sweety439 No, it just have to > all prime factors of gcd(k+-1,b-1), thus we can sieve start with p=5 (...)
"We can" but srsieve won't. The error message when trying to sieve from P0<=43 for 13*43^n-1 is:
Code:
ERROR: Sieve range P0 <= p <= P1 must be in 43 < P0 < P1 < 2^62.
That's why you have to change the header of the sieve first before sieving. Try it.

2020-07-21, 15:01   #27
sweety439

Nov 2016

22×3×197 Posts

Quote:
 Originally Posted by kar_bon "We can" but srsieve won't. The error message when trying to sieve from P0<=43 for 13*43^n-1 is: Code: ERROR: Sieve range P0 <= p <= P1 must be in 43 < P0 < P1 < 2^62. That's why you have to change the header of the sieve first before sieving. Try it.
okay, I used srsieve for R36, sieved k*36^n-1 start with p=11 (to p=10^8) for these 193 k's (which remain at n=1000): {251, 260, 924, 1148, 1356, 1555, 1923, 2110, 2133, 2443, 2753, 2776, 3181, 3590, 3699, 3826, 3942, 4241, 4330, 4551, 4635, 4737, 4865, 5027, 5196, 5339, 5483, 5581, 5615, 5791, 5853, 6069, 6236, 6542, 6581, 6873, 6883, 7101, 7253, 7316, 7362, 7399, 7445, 7617, 7631, 7991, 8250, 8259, 8321, 8361, 8363, 8472, 8696, 9140, 9156, 9201, 9469, 9491, 9582, 10695, 10913, 11010, 11014, 11143, 11212, 11216, 11434, 11568, 11904, 12174, 12320, 12653, 12731, 12766, 13641, 13800, 14191, 14358, 14503, 14540, 14799, 14836, 14973, 14974, 15228, 15578, 15656, 15687, 15756, 15909, 16168, 16908, 17013, 17107, 17354, 17502, 17648, 17749, 17881, 17946, 18203, 18342, 18945, 19035, 19315, 19389, 19572, 19646, 19907, 20092, 20186, 20279, 20485, 20630, 20684, 21162, 21415, 21880, 22164, 22312, 22793, 23013, 23126, 23182, 23213, 23441, 23482, 23607, 23621, 23792, 23901, 23906, 23975, 24125, 24236, 24382, 24556, 24645, 24731, 24887, 24971, 25011, 25052, 25159, 25161, 25204, 25679, 25788, 25831, 26107, 26160, 26355, 26382, 26530, 26900, 27161, 27262, 27296, 27342, 27680, 27901, 28416, 28846, 28897, 29199, 29266, 29453, 29741, 29748, 29847, 30031, 30161, 30970, 31005, 31190, 31326, 31414, 31634, 31673, 31955, 32154, 32302, 32380, 32411, 32451, 32522, 32668, 32811, 33047, 33516, 33627, 33686, 33762} and for 1001<=n<=10^5, then changed the first row of the t17_b36 file to "ABC ($a*36^$b-1)/gcd(\$a-1,35)", then used pfgw to test the primality of these numbers.

 2020-07-21, 15:17 #28 sweety439     Nov 2016 22×3×197 Posts currently R36 (with CK=33791) at n=10K, with these k's remain: {1148, 1555, 2110, 2133, 3699, 4551, 4737, 6236, 6883, 7253, 7362, 7399, 7991, 8250, 8361, 8363, 8472, 9491, 9582, 11014, 12320, 12653, 13641, 14358, 14540, 14836, 14973, 14974, 15228, 15687, 15756, 15909, 16168, 17354, 17502, 17946, 18203, 19035, 19646, 20092, 20186, 20630, 21880, 22164, 22312, 23213, 23901, 23906, 24236, 24382, 24645, 24731, 24887, 25011, 25159, 25161, 25204, 25679, 25788, 26160, 26355, 27161, 29453, 29847, 30970, 31005, 31634, 32302, 33047, 33627}, I know that square k's proven composite by full algebra factors (except k=1, since for n=2, (1*36^2-1)/gcd(1-1,36-1) = 37 is prime, however k=1 can only have this prime and cannot have no more primes (thus cannot have infinitely many primes), thus k=1 is still excluded from the conjecture (see post https://mersenneforum.org/showpost.p...&postcount=315 for more information), a k-value is included from the conjecture if and only if this k-value may have infinitely many primes; also, I know that the k's such that gcd(k-1,36-1) = 1 is completely the same as the R36 problem in CRUS for these k's, however I don't have the primes for these k's other than the top 10 primes in CRUS (since gcd(33791-1,36-1) is not 1, thus the CK for the CRUS R36 conjecture cannot be 33791 (it is 116364)), thus I only listed the (probable) primes for n<=10K (in which I have searched) in the file, the (probable) primes for 1K
2020-07-25, 20:16   #29
sweety439

Nov 2016

22·3·197 Posts

Found the conjectured smallest Sierpinski/Riesel numbers for bases <= 2500

* Only the k's with covering set are considered as Sierpinski/Riesel numbers, k-values that make a full covering set with all or partial algebraic factors are excluded from the conjectures.
* Searched to k=5M, listed "NA" if the conjectured smallest Sierpinski/Riesel number for this base is >5M (i.e. there is no k <= 5M with covering set)
* Test limit: primes in the covering set <= 100K, exponents <= 2100
Attached Files
 Conjectured smallest Sierpinski.txt (22.0 KB, 20 views) Conjectured smallest Riesel.txt (22.0 KB, 21 views)

 2020-09-17, 16:55 #30 sweety439     Nov 2016 93C16 Posts The mixed Sierpinski/Riesel conjectures for prime bases This project is from the article http://www.kurims.kyoto-u.ac.jp/EMIS...rs/i61/i61.pdf, this article is about the mixed Sierpinski (base 2) theorem, which is that for every odd k<78557, there is a prime either of the form k*2^n+1 or of the form 2^n+k, we generalized this theorem (may be only conjectures to other bases) to other prime bases (since the dual form for composite bases is more complex when gcd(k,b) > 1 (see thread https://mersenneforum.org/showthread.php?t=21954), we only consider prime bases), we conjectured that for every k
 2020-09-17, 17:12 #31 sweety439     Nov 2016 22·3·197 Posts Also, 31^5-55758 is prime, thus the mixed Riesel conjecture base 31 is also a theorem. For the S37 case, 37^1+94, 37^5+1272, and 37^2+2224 are primes, thus the mixed Sierpinski conjecture base 37 is also a theorem. For the S43 case, 43^n+166 is composite for all small n, thus the mixed Sierpinski conjecture base 43 is still unsolved. For the R23 case, 23^568-404 is prime, thus the mixed Riesel conjecture base 23 is also a theorem. R37 case has 3 k-values remain: 1578, 6752, and 7352: Code: |37^3-522| |37^30-816| |37^4-1614| |37^1-2148| |37^298-2640| |37^3-3972| |37^1-4428| |37^401-5910| |37^33-7088|

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