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For integer b2, We define:

  • Sierpinski number base b: Positive integer k such that gcd(k+1,b1)=1 and kbn+1 is composite for all integer n1
  • Riesel number base b: Positive integer k such that gcd(k1,b1)=1 and kbn1 is composite for all integer n1
  • Dual Sierpinski number base b: Positive integer k such that gcd(k,b)=1 and gcd(k+1,b1)=1 and bn+k is composite for all integer n1
  • Dual Riesel number base b: Positive integer k such that gcd(k,b)=1 and gcd(k1,b1)=1 and bnk is composite for all integer n1 such that bn>k+1

See http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm for the (conjectured) smallest Sierpinski number base b, and see http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm for the (conjectured) smallest Riesel number base b

If a number k is Sierpinski number base b and gcd(k,b)=1, then k is also a dual Sierpinski number base b, and if a number k is Riesel number base b and gcd(k,b)=1, then k is also a dual Riesel number base b, since they have the same covering set, the exception is when the prime itself is in the covering set, e.g. 11 is a Riesel number base 14 but not dual Riesel number base 14, since its covering set is {3,5}, but 14111 is exactly 3 and the smallest dual Riesel number base 14 is 19, also, 13 is a Riesel number base 20 but not dual Riesel number base 20, since its covering set is {3,7}, but 20111 is exactly 7 and the smallest dual Riesel number base 20 is 29

For b=3, since the (conjectured) smallest Sierpinski number 125050976086 and the (conjectured) smallest Riesel number 63064644938 are coprime to 3, they are also the (conjectured) smallest dual Sierpinski number and the (conjectured) smallest dual Riesel number, respectively.

For b=6, since neither the (conjectured) smallest Sierpinski number 174308 nor the (conjectured) smallest Riesel number 84687 are coprime to 6, I continue to search and found the (conjectured) smallest dual Sierpinski number 282001 and the (conjectured) smallest dual Riesel number 333845.

For b=7, since the (conjectured) smallest Sierpinski number 1112646039348 and the (conjectured) smallest Riesel number 408034255082 are coprime to 7, they are also the (conjectured) smallest dual Sierpinski number and the (conjectured) smallest dual Riesel number, respectively.

For b=15, neither the (conjectured) smallest Sierpinski number 91218919470156 nor the (conjectured) smallest Riesel number 36370321851498 are coprime to 15, and they are too large to search. My question is:

What are the (conjectured) smallest dual Sierpinski number and the (conjectured) smallest dual Riesel number in base b=15?

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  • If I'm not mistaken, the factorization of (15301)/(257) gives a covering set of primes, so that should give an order-of-magnitude estimate for an upper bound, and some Chinese Remainder Thm work would then give actual dual S- and R-numbers. Commented 17 hours ago
  • Not sure if (15241)/(277) provides a covering set -- lets work it out. 15 is order 3 mod 241, order 4 mod 113, order 6 mod 211. That gives ~9/12 coverage. It is order 8 mod 17 and 1489, order 12 mod 13 and 3877, and order 24 mod a 10-digit prime. So we can drop a congruence mod 3877, and we're left with a congruence mod 13*17*113*211*241*1489 ~2*10^13. So that's my current best guess on the size of the minimal dual Sierpinski and Riesl numbers. Commented 17 hours ago
  • Sorry, we can't drop the 3877, but i think we can drop the 10-digit prime. So 8*10^16? You lose 1/12 coverage from the order-3 and order-4 cycle overlaps, and another 2/24 from the order-3 and order-8 overlaps. You need both order-12 primes to compensate. Commented 16 hours ago

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