Problems & Puzzles: Problems

Problem 68. More on Brier numbers.

Arkadiusz Wesolowski sent the following puzzle proposal, related to Brier numbers. Regarding this issue, see previous Problems 29, 49, 52 & 58.

A Brier number is an integer k such that both k.2n+1 and k.2n-1 are composite for any value of n.

Let k be a Brier number. Let A be a covering set for the Sierpinski number k and let B be a covering set for the Riesel number k, then A ∪ B = S = {p(1), p(2), ..., p(s)} a set of primes and P = product[p(i)](i = 1...s).
 
Number of primes used in both sets A & B Discoverer P value
1 Cohen & Selfridge (See 1974) 33241542480794255062005345795
2 Arkadiusz Wesolowski (June, 2017) 22366096283725870121540301161955
3 ? ?
4 ? ?


Example by Cohen & Selfridge:

A={3, 5, 13, 17, 97, 257, 241}
B={3, 7, 11, 19, 31, 37, 41, 61, 73, 109, 151, 331}
P=3*5*13*17*97*257*241*7*11*19*31*37*41*61*73*109*151*331=
33241542480794255062005345795 (29 digits)
k=47867742232066880047611079

Example by Arkadiusz:

A = {3, 5, 13, 17, 19, 241, 433, 38737}
B = {3, 7, 11, 19, 31, 37, 41, 61, 73, 109, 151, 331}
P = 3*5*13*17*19*241*433*38737*7*11*31*37*41*61*73* 109*151*331 = 22366096283725870121540301161955 (32 digits)
k=2769569390938829068824114667
Q. Can you find smallest values of P for which "Number of primes used in both sets (A and B)" = 1, 2, 3, 4, etc

Arkadiusz Wesolowski wrote on August 16, 2017_

I have found a new solution.
 
The number of primes used in both sets (A and B) is 2.
 
A = {3, 5, 17, 19, 97, 109, 241, 257, 433}
B = {3, 7, 11, 13, 19, 31, 37, 41, 61, 73, 151, 331}
P = 3*5*17*19*97*109*241*257*433*7*11*13*31*37*41*61*73*151*331 = 
14393587894183912441848314729235 (32 digits)
k = 1631535784434767401179758185111

***


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