other conjectures

[sweety439]

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

[gd_barnes]

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.

[sweety439]

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)

[kar_bon]

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:

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.

[sweety439]

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:

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.

[sweety439]

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<n<=10K are in post https://mersenneforum.org/showpost.p...&postcount=779

[sweety439]

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

[sweety439]

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<the CK for the Sierpinski conjecture base b (for prime b) which is not divisible by b, there is a prime either of the form k*b^n+1 or of the form b^n+k

We can also generalize this problem to the Riesel side, for the classic (base 2) mixed Riesel problem, there is only 7 unsolved k-values: 2293, 196597, 304207, 342847, 344759, 386801, 444637 (and plus this 2 k-values if probable primes cannot be consider as primes: 363343 and 384539) (see thread https://mersenneforum.org/showthread.php?t=6545), we conjectured that for every k<the CK for the Riesel conjecture base b (for prime b) which is not divisible by b, there is a prime either of the form k*b^n-1 or of the form |b^n-k|

Note that the weight of b^n+k is the same as that of k*b^n+1, and the weight of |b^n-k| is the same as that of k*b^n-1, if gcd(k,b)=1

S3 and S7 have too many k's remain, for S5, we have these primes:

5^24+6436
5^36+7528
5^144+10918
5^1505+26798
5^4+29914
5^458+36412
5^3+41738
5^9+44348
5^485+44738
5^12+45748
5^12+51208
5^46+58642
5^12+60394
5^2+62698
5^2+64258
5^10+67612
5^41+67748
5^13+71492
5^74+74632
5^7+76724
5^3+83936
5^21+84284
5^181+90056
5^23+92906
5^4+93484
5^11+105464
5^11+126134
5^1+139196
5^15+152588

thus the only mixed-remain k-value is 31712

S11 and S13 are already proven, for S17, 17^838+244 is prime, thus, the mixed Sierpinski conjecture base 17 is also a theorem.

For the Riesel side, R3 and R7 also have too many k remain, for R5, we have these primes:

|5^1-3622|
|5^11-4906|
|5^920-23906|
|5^6-26222|
|5^199-35248|
|5^12-52922|
|5^9-63838|
|5^6-64598|
|5^695-71146|
|5^35-76354|
|5^24-109862|
|5^65-127174|
|5^27-131848|
...

the remain k-values are 68132, 81134, 102952, 109238, 134266, ...

R11, R13, and R17 are already proven.

[sweety439]

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:

|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|

[sweety439]

Originally Posted by sweety439

for S5, we have these primes:

5^24+6436
5^36+7528
5^144+10918
5^1505+26798
5^4+29914
5^458+36412
5^3+41738
5^9+44348
5^485+44738
5^12+45748
5^12+51208
5^46+58642
5^12+60394
5^2+62698
5^2+64258
5^10+67612
5^41+67748
5^13+71492
5^74+74632
5^7+76724
5^3+83936
5^21+84284
5^181+90056
5^23+92906
5^4+93484
5^11+105464
5^11+126134
5^1+139196
5^15+152588

thus the only mixed-remain k-value is 31712

5^50669+31712 is PRP!!! (35417 digits, to small for the PRP top)

Thus, the "mixed Sierpinski conjecture base 5" is now a theorem, in the weak case that probable primes can be regard as proven primes.