A counterexample to a conjecture of Thakur on Carlitz-Wieferich primes

Abstract

Let A = F_q[T] with q a power of an odd prime p, let [n] = T^(q^n) - T, and let rho be the Carlitz module. A monic prime P of A is a c-Wieferich prime (to base 1) if rho_P(1) = 1 mod P^2. Thakur suggested in 2015, on the basis of limited data and of proofs in degrees 2 and 3, that in odd characteristic every c-Wieferich prime has degree divisible by p; the question was restated as open in 2024, and Bamunoba and Bergstrom, after extensive computations, expressed the belief that the statement holds in odd characteristic. We show that it is false: an explicit irreducible c-Wieferich prime of degree 5 over F_{19^3} is exhibited, with 19 not dividing 5. We further give a closed form for the resulting common factor of [5] and M_5: it equals mu(T^q - T) for an explicit quintic mu with coefficients in the prime field F_19, squarefree of degree 5*19^3, and it divides gcd([5], M_5); we conjecture equality, and verify it for the part of low degree over the prime field. Degree 5 is the least possible degree of such a counterexample, and exhaustive computations show that no counterexample exists over the prime fields F_p in a substantial range of degrees and characteristics. The proof that degree 5 is minimal, and the method by which the example was found, appear in a companion paper.