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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.

Exact FrontierDelta

Prior state unknown→disproved

Scope and record

Occurred: Jul 14, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: site-confirmed. Publication: preprint. AI contribution: ai-discovered. Imported under CC BY 4.0.

Canonical aliases: Thakur's Conjecture on Carlitz-Wieferich Primes · Carlitz-Wieferich primes

Confidence: Not scored

Registry verification: site confirmed · preprint · resolved

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Attribution

VibeMathed
registry · event recorded by

David Niedbala Giraudin
human · human collaborator

Claude Opus 4.8
model · ai model contributor · Anthropic

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

VibeMathed record: Thakur's Conjecture on Carlitz-Wieferich Primes

specialist registry · passed

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 arXiv preprint (v2), not peer-reviewed.

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Thakur's Conjecture on Carlitz-Wieferich Primes

preprint · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

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Lineage and corrections

This event attributed to David Niedbala Giraudin

This event attributed to Claude Opus 4.8

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