Let A be the coefficient ring and let phi:A rightarrow Atau be a Drinfeld module. For each prime polynomial P, let g_P,phi denote the auxiliary polynomial associated with P and phi. Prime-values conjecture. There exist infinitely many prime polynomials P such that g_P,phi is prime. This conjecture is stated as the remaining assumption needed to prove that infinitely many values phi_P(a) are non-prime, or equivalently that infinitely many phi-Mersenne numbers are not prime; the supplied text gives no resolution.

The prime-values conjecture for Drinfeld-module auxiliary polynomials

The conjecture asks whether every Drinfeld module has infinitely many prime polynomials $P$ for which the associated auxiliary polynomial $g_{P,\phi}$ is prime. It is used as the remaining assumption in a result on infinitely many non-prime Drinfeld-module Mersenne numbers.

Known results

For the Carlitz module, $g_{P,\phi}=P-1$ , and Hall’s theorem gives infinitely many prime polynomials $P$ for which both $P$ and $P-1$ are prime.

December 2025 preprint

Alexis Lucas’s Wieferich and Mersenne primes for function fields states the conjecture and proves consequences assuming it, but reports no proof or counterexample for the general case.

Current status (as of September 2026): The conjecture remains open for general Drinfeld modules; it is known for the Carlitz module, and no proof or counterexample for the general statement has been reported.

The prime-values conjecture for Drinfeld-module auxiliary polynomials

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Progress summary

Solutions 0