Liskovets-Gallot conjectures

The Liskovets-Gallot conjectures are a family of conjectures regarding the frequency of prime n-values of a given parity for Riesel and Proth k-values divisible by 3. The notion that certain k-values, divisible by 3, have no prime n-values of a given parity was first conjectured by Valery Liskovets in 2001. Yves Gallot found suitable non-trivial k-values for all four sign/parity combinations shortly thereafter and claimed them to be the smallest such k-values, thus forming the conjectures' final form. A search was started by CRUS in 2008 to prove the conjectures by finding primes of the required parity for all smaller k-values. The even Proth conjecture was proven in 2015, and CRUS is continuing the CRUS Liskovets-Gallot subproject to find the remaining 9 primes required to prove the other 3 conjectures.

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 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

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.

Search

The current search is maintained by the Conjectures 'R Us project and can be found here.

Links

• Problem 36 "The Liskovets-Gallot numbers" from PP&P connection by Carlos Rivera