Breakthrough Tracker record
A counterexample to Thakur's conjecture on Carlitz–Wieferich primes
The paper gives an explicit degree-five Carlitz–Wieferich prime over the field with 19 cubed elements. Because 19 does not divide five, the example contradicts the conjectured degree-divisibility condition.
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- Stable ID
math-thakur-carlitz-wieferich-counterexample-2026- Revision
math-thakur-carlitz-wieferich-counterexample-2026.v1- Field
- Mathematics · Function-field number theory
- Evidence
- Tier 1 · Peer reviewed: No
- Record state
- Provisional · Preprint counterexample; reported independent check
- Last checked
AI role
No substantive AI role was disclosed in the inspected manuscript.
Record details
- Problem or result
- Thakur's conjectural degree-divisibility condition for Carlitz–Wieferich primes in odd characteristic
- Authors
- David Niedbala Giraudin
- Institutions
- Independent researcher, Meaux, France
- Result date
- Version 2 submitted July 22, 2026
Why it matters
It overturns a conjectural pattern in function-field arithmetic that had survived earlier proofs in small degrees and extensive computations.
Limits
Version 2 corrected an unrelated exactness theorem to a conjecture. The counterexample remains in the manuscript, and the author reports an independent Magma check by Dinesh Thakur, but no separate verification artifact or journal review was located.
Sources
Correction and revision history
- 2026-07-22 — Added from version 2. The entry discloses that former Theorem 4.2 was corrected from an exactness statement to a conjecture; the counterexample itself remains.
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