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Liskovets-Gallot conjectures

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Definitions

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
51 38 5
231 51 9
261 56 14
87 2 36
93 1 38
177 8 46

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 n=even: k=66741
k•2n+1 is composite for all n=odd: k=95283
k•2n-1 is composite for all n=even: k=39939
k•2n-1 is composite for all n=odd: k=172677

Proof

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.

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