Valery Liskovets studied the list of k•2n+1 primes and observed, that the k's (k divisible by 3) got an irregular contribution of odd and even exponents yielding a prime.
Examples: (for 1 ≤ n ≤ 100000)
|k-value||# odd||# even|
So Liskovets formulated the conjecture:
There exist k, 3|k, such that primes k•2n+1 do exist but only with odd n/only with even n.
Yves Gallot extended this for k•2n-1 numbers and gave also the first solutions as:
- k•2n+1 is composite for all even n for k=66741
- k•2n+1 is composite for all odd n for k=95283
- k•2n-1 is composite for all even n for k=39939
- k•2n-1 is composite for all odd n for k=172677
The verification of these conjectures has to be done in the same manner like the Riesel problem: find a prime for all k-values less than the given with the needed condition.