What was already known
Thakur had ruled out degrees two and three and suggested the divisibility rule. Earlier odd-characteristic computations found only fixed examples; characteristic-two counterexamples were already known.
AI involvement
Claude Opus 4.8 helped formulate the problem and designed and ran exact computations under Niedbala Giraudin’s direction. Thakur independently checked the example and suggested simplifications.
The assistant supplied the mathematical formalization and all computations for the retained counterexample, while the human set direction and contributed structural reading.
Importance
The explicit degree-five example refutes Thakur’s proposed rule in odd characteristic.
Limitations
The proposed exact description of all 6859 translates in a related gcd remains a conjecture. Thakur independently checked the counterexample in Magma; the companion account of the discovery method is pending.
Technical verification
The precise finite-field certificate—irreducibility and the Carlitz–Wieferich congruence—was checked with exact arithmetic in independent systems, including Thakur's independent Magma calculation.
Formal checking
Independent exact finite-field computations; Thakur used Magma
- Checker
- Niedbala Giraudin and Dinesh Thakur
- Scope
- The explicit degree-five polynomial is irreducible and Carlitz–Wieferich over F_(19^3)
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Thakur explains shortcuts in verifying the example and its translation class, but the paper says the author cannot verify the mathematics directly; a human explanation of the full construction strategy is not documented.
Correction and publication history
The explicit degree-five example refutes Thakur’s proposed rule in odd characteristic.
Read this published revision