 mersenneforum.org A Sierpinski/Riesel-like problem
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sweety439

"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

3·29·41 Posts The correct text file for extended R17 is here. (the above file only lists the k's < 44, we should lists all k's < 49)
Attached Files extend-Riesel-base17.txt (299 Bytes, 226 views)   2016-12-24, 19:45 #35 Batalov   "Serge" Mar 2008 Phi(4,2^7658614+1)/2 100110101011012 Posts You can surely post 10 messages an hour. You have proven your capacity to type. You come across as a graphomaniac. But do you think anybody is reading your inane stream of conscience? Why would they -- if you are appear to be deaf to other people's messages?   2016-12-26, 19:26 #36 sweety439   "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 3·29·41 Posts The (probable) primes with n > 1000 for the extended Sierpinski/Riesel problems (with bases b <= 32, except b = 2, 3, 6, 15, 22, 24, 28, 30) are: S4: 186*4^10458+1 S7: (141*7^1044+1)/2 S10: 804*10^5470+1 S12: 404*12^714558+1 378*12^2388+1 S16: (23*16^1074+1)/3 S17: 10*17^1356+1 S18: 122*18^292318+1 381*18^24108+1 291*18^2415+1 S25: (61*25^3104+1)/2 S26: 32*26^318071+1 217*26^11454+1 95*26^1683+1 178*26^1154+1 R4: (106*4^4553-1)/3 74*4^1276-1 296*4^1275-1 ( = 74*4^1276-1, the two primes are the same, since 296 = 74 * 4) R7: (367*7^15118-1)/6 (313*7^5907-1)/6 (159*7^4896-1)/2 (429*7^3815-1)/2 (419*7^1052-1)/2 R12: (298*12^1676-1)/11 R17: 44*17^6488-1 (29*17^4904-1)/4 (13*17^1123-1)/4 R25: 86*25^1029-1 R26: 115*26^520277-1 32*26^9812-1 (121*26^1509-1)/5 Last fiddled with by sweety439 on 2017-02-07 at 15:02   2016-12-26, 19:34   #37
sweety439

"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

3·29·41 Posts Quote:
 Originally Posted by sweety439 Extend Sierpinski problem base b: Finding and proving smallest k such that (k*b^n+1)/gcd(k+1, b-1) is composite for all integer n>1. Extend Riesel problem base b: Finding and proving smallest k such that (k*b^n-1)/gcd(k-1, b-1) is composite for all integer n>1.
It should not be "n>1", it should be "n>=1", we allow n=1 for (k*b^n+1)/gcd(k+1, b-1) and (k*b^n-1)/gcd(k-1, b-1), but we do not allow n=0.

Last fiddled with by sweety439 on 2016-12-26 at 19:35   2016-12-26, 19:41   #38
sweety439

"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

3·29·41 Posts Quote:
 Originally Posted by gd_barnes First suggestion: When showing which k's need a prime use the k-value in this form: k*3^n+(3^n-1)/2 Reasoning for the suggestion. For a more complex base/append combination, such as base 9 append 7, it would not be clear what k is being searched unless you use the form: k*9^n+(9^n-1)*7/8 Second suggestion: Do not show k=3m as remaining. It is redundant.
This is another problem, different from the "Add repeated digits after a number until it is prime" problem. First, although for the Riesel case this problem are special cases (when d divides b-1) of that problem, but for the Sierpinski case it is not. Second, in that problem, if d does not divide b-1, then it cannot be transfer to this problem.

Thus, in this problem, we say "k=1617 is remaining for the extended Riesel base 3 problem", but in that problem, we say "k=806 is remaining for the base 3 d=1 problem".   2016-12-27, 16:48 #39 sweety439   "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 3×29×41 Posts These problems are just my extending for the original Sierpinski/Riesel problems. Last fiddled with by sweety439 on 2016-12-27 at 16:48   2016-12-27, 16:49 #40 sweety439   "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 3·29·41 Posts For (269*10^n+1)/9, tested to n=6000, still no (probable) prime found.   2016-12-28, 16:34 #41 sweety439   "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 3·29·41 Posts For the two bases that have only one k remaining: S10, k=269: (269*10^n+1)/9 seems to be tested to n=10000 (I ran the program of 25 hours and 38 minutes!!!), still no (probable) prime found, base released. R7, k=197: See the link: https://www.rose-hulman.edu/~rickert/Compositeseq/, (197*7^n-1)/2 is already tested to n=15000 with no (probable) prime found. Last fiddled with by sweety439 on 2016-12-28 at 16:35   2016-12-28, 16:43   #42
sweety439

"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

3×29×41 Posts Also tested extended S15, S22 and S24 to k=500.

For extended S15, there are 4 k's <= 500 remain: 219, 225, 341, 343.
For extended S22, there are 5 k's <= 500 (excluding the GFNs: i.e. k=22 and k=484) remain: 173, 346, 383, 461, 464.
For extended S24, there are 5 k's <= 500 remain: 319, 346, 381, 461, 486. However, all of them are correspond to CRUS primes, since all these k's satisfy that gcd(k+1,24-1)=1, and according to the CRUS page, the smallest remain k for the original S24 problem is k=656. Thus, there are in fact no k's <= 500 remain for extended S24.
Attached Files extend-Sierp-base15.txt (3.4 KB, 205 views) extend-Sierp-base22.txt (3.3 KB, 211 views) extend-Sierp-base24.txt (3.4 KB, 206 views)

Last fiddled with by sweety439 on 2016-12-28 at 17:26   2016-12-28, 16:51 #43 sweety439   "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 356710 Posts If the k satisfies that gcd(k+-1, b-1)=1, then the correspond prime of the extended Sierpinski/Riesel problem is the same as the correspond prime of the original Sierpinski/Riesel problem. The extended Sierpinski/Riesel problem is just my extending of the original Sierpinski/Riesel problem to the k's such that gcd(k+-1, b-1) is not 1. Since k*b^n+-1 is always divisible by gcd(k+-1, b-1), the division is necessary for there to be any chance of finding primes. Last fiddled with by sweety439 on 2016-12-28 at 16:53   2016-12-28, 17:20   #44
sweety439

"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

3×29×41 Posts Also tested extended R15, R22 and R24 to k=500.

For extended R15, there are 6 k's <= 500 remain: 47, 203, 239, 407, 437, 451.
For extended R22, there are 4 k's <= 500 (k=185 prime is given by the CRUS page) remain: 208, 211, 355, 436.
For extended R24, there are 7 k's <= 500 remain: 69, 201, 339, 346, 364, 389, 461. However, the k's != 1 mod 23 are correspond to CRUS primes, according to the CRUS page, k=389 is the only k <= 500 remain for the original R24 problem.
Attached Files extend-Riesel-base15.txt (3.3 KB, 207 views) extend-Riesel-base22.txt (3.4 KB, 199 views) extend-Riesel-base24.txt (3.4 KB, 212 views)

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