Degree restrictions for Carlitz–Wieferich primes

Must the degree of a Carlitz–Wieferich prime in odd characteristic be divisible by that characteristic? Thakur’s proposed rule concerns these special primes in polynomial arithmetic; characteristic-two counterexamples do not settle it.

Why it matters and what counts as progress

Why it matters: An odd-characteristic exception tests a degree restriction that characteristic-two examples cannot decide.

What counts as progress: Prove the rule in a new odd-characteristic range or give an explicit irreducible prime of degree not divisible by the characteristic.

How the problem developed

2015 · Conjecture [1]
Dinesh Thakur Suggested that in odd characteristic every base-one c-Wieferich prime has degree divisible by the characteristic, based on limited data and proofs in degrees two and three.
14 Jul 2026 · Counterexample [2]
David Niedbala Giraudin Exhibited a degree-five c-Wieferich prime over F_(19^3), disproving the divisibility conjecture since 19 does not divide 5; D. Thakur independently checked the example in Magma.

AI contribution

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

David Niedbala Giraudin

Niedbala Giraudin gives an explicit irreducible degree-five Carlitz–Wieferich prime over F_(19³), refuting the proposed odd-characteristic rule that its degree must be divisible by the characteristic.

AI involvementAI-led

ImportanceNotable

Technical verificationFormal (exact computation in SageMath, PARI/GP, FLINT, galois and Magma)

Human understandingPartial