Thakur's Conjecture on Carlitz-Wieferich Primes

A monic prime $P$ of $\mathbb{F}_q[T]$ is a $c$ -Wieferich prime if $\rho_P(1) \equiv 1 \bmod P^2$ for the Carlitz module $\rho$ . On limited data and proofs in degrees $2$ and $3$ , Thakur suggested in 2015 that in odd characteristic every $c$ -Wieferich prime has degree divisible by $p$ . It is false: an explicit irreducible $c$ -Wieferich prime has degree not divisible by $p$ , and the resulting common factor has a closed form.

Result: Disproved
Status: Resolved
AI contribution: AI-discovered
Method: Construction
Field: Function field arithmetic
Posed by: Dinesh Thakur
Year posed: 2015
Years open: 11y
Solved: 2026-07-14
Model: Claude Opus 4.8
Vendor: Anthropic
Collaborators: David Niedbala Giraudin
Verification: Site-confirmed
Publication: Preprint
Significance: 10 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What the AI did

The methodology section is the most explicit division of labour in this batch. The author is an independent researcher with no formal mathematical training. He set the research direction and the criteria for which problems to pursue and contributed a structural, visual reading of the objects; the model proposed problems meeting those criteria and supplied the mathematical domain knowledge, the formalization, the drafting, and the design and execution of all computations, under his direction. The strategy emerged from the dialogue. Lacking the training to verify the mathematics directly, the author relied on exact computational checks reproduced across independent systems.

Verification

Independently reproduced. We recomputed the claim from the definitions in arXiv:2607.15305, from scratch and with no computer-algebra dependency, so the check shares no code with the author's appendix. Confirmed: $x^3-8x^2-4x-11$ is irreducible over $\mathbb{F}_{19}$ , so $\mathbb{F}_{19^3}$ is a field; $P$ is monic of degree $5$ , irreducible over $\mathbb{F}_{19^3}$ , and genuinely uses the cubic extension; and $\rho_P(1) \equiv 1 \bmod P^2$ , which is the definition of a $c$ -Wieferich prime, computed through the Carlitz recursion inside $\mathbb{F}_q[T]/(P^2)$ . The Bamunoba-Bergstrom criterion the paper cites, $M_5(\theta) = 0$ , was computed by a separate route and agrees. We also confirmed that $\mu(X)$ divides $X + X^q + \cdots + X^{q^4}$ , which is what makes $G = \mu(T^q - T)$ divide $[5]$ . Since $19 \nmid 5$ , the counterexample stands. This matters more than usual here because the author states he cannot verify the mathematics directly. arXiv preprint (v2), not peer-reviewed.

Source

arXiv:2607.15305 - A counterexample to a conjecture of Thakur on Carlitz-Wieferich primes

Thakur's Conjecture on Carlitz-Wieferich Primes

What the AI did

Verification

Source

Discussion