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

Technical Summary: A Counterexample to a Conjecture of Thakur on Carlitz–Wieferich Primes

Problem Statement
The paper addresses a conjecture regarding Carlitz–Wieferich primes in the context of function fields over finite fields of odd characteristic. Let $p$ be an odd prime, $q = p^k$, and $A = \mathbb{F}_q[T]$. A monic prime $P \in A$ is defined as a c-Wieferich prime (to base 1) if it satisfies the congruence $\rho_P(1) \equiv 1 \pmod{P^2}$, where $\rho$ is the Carlitz module. This condition is the function field analogue of the classical Wieferich prime condition $2^{p-1} \equiv 1 \pmod{p^2}$.

In 2015, Dinesh Thakur, based on proofs for degrees 2 and 3 and limited numerical data, conjectured that in odd characteristic, every c-Wieferich prime must have a degree $d$ divisible by $p$. This conjecture was restated as an open problem in 2024, and subsequent extensive computations by Bamunoba and Bergström supported the belief that the statement holds true for odd characteristics. The paper aims to test the validity of this conjecture.

Methodology
The author, an independent researcher, employed a systematic computational method developed in a companion paper [4] to search for counterexamples. The methodology involves:

1. Reduction to Finite Computation: The existence of a c-Wieferich prime of a prescribed degree $d$ in a prescribed odd characteristic is reduced to a finite, effective computation involving trace-zero elements in extension fields.
2. Exact Arithmetic: All computations were performed using exact arithmetic (no floating-point approximations) in standard public computer-algebra systems (SageMath, PARI/GP, FLINT, and the galois Python library).
3. Verification: Results were reproduced on independent engines and calibrated against known c-Wieferich primes (e.g., in $\mathbb{F}_3[T]$ and $\mathbb{F}_5[T]$) and ordinary primes.
4. Collaborative Workflow: The research direction and structural interpretation were provided by the author, while the AI assistant (Claude Opus 4.8) handled the mathematical domain knowledge, formalization, drafting, and execution of computations.

Key Contributions and Results

1. Disproof of the Conjecture:
   The paper provides a definitive counterexample to Thakur's conjecture in odd characteristic. The authors construct an irreducible c-Wieferich prime $P(T)$ of degree 5 over the field $\mathbb{F}_{19^3} = \mathbb{F}_{19}[c]$ (where $c^3 = 8c^2 + 4c + 11$).
   The polynomial is:
   $$P(T) = T^5 + (11 + 17c + 9c^2)T^4 + (3 + 7c + 18c^2)T^3 + (2 + 5c + 6c^2)T^2 + (3 + 3c + 11c^2)T + (6 + 17c + 5c^2)$$
   Crucially, the degree $d=5$ is not divisible by the characteristic $p=19$ (i.e., $19 \nmid 5$), directly contradicting the conjecture.

2. Closed Form for the Common Factor:
   The paper derives a closed form for the greatest common divisor of the polynomials $[5]$ (the product of all monic primes of degree dividing 5) and $M_5$ (a specific polynomial related to the c-Wieferich condition).
   Let $q = 19^3$. The polynomial $G = \mu(T^q - T)$, where $\mu(X) = X^5 + 5X^3 + 3X^2 - 4X - 9 \in \mathbb{F}_{19}[X]$, is shown to be:

   • Squarefree of degree $5 \cdot 19^3 = 34,295$.
   • The product of exactly 6,859 distinct c-Wieferich primes of degree 5 (the translation class of $P(T)$).
   • Equal to $\gcd([5], M_5)$ in $\mathbb{F}_{19^3}[T]$.

3. Minimality of the Degree:
   The paper establishes that degree 5 is the minimal possible degree for such a counterexample in odd characteristic:

   • Degrees 1, 2, and 3 are excluded by prior results (Thakur [6]).
   • Degree 4 is proven to have no c-Wieferich primes in any odd characteristic (Theorem 5.1, proved in the companion paper [4]).

4. Non-Existence over Prime Fields (in a Substantial Range):
   The authors performed exhaustive computations to determine if counterexamples exist over prime fields $\mathbb{F}_p$ (where $q=p$). Proposition 5.2 reports that no c-Wieferich primes of degree $d$ exist over $\mathbb{F}_p$ for $p \nmid d$ within a substantial range of degrees and characteristics (e.g., degree 5 for $p \in \{3, 7, 11, \dots, 37\}$). The known counterexample requires a cubic extension ($q=19^3$), not a prime field.

Significance and Claims
The paper claims to resolve the open question regarding the divisibility of the degree of c-Wieferich primes by the characteristic in odd settings. It demonstrates that the conjecture is false, even in odd characteristic, and that the phenomenon is not restricted to characteristic 2.

The significance of the result is twofold:

1. Refutation of a Stronger Expectation: The counterexample is "non-fixed," meaning it is not invariant under additive translation ($P(T) \neq P(T+\gamma)$). Previous computational searches in odd characteristic had only found "fixed" examples, which necessarily have degrees divisible by $p$. This result answers negatively the stronger expectation that no non-fixed c-Wieferich primes exist in odd characteristic.
2. Structural Insight: The discovery of a closed-form expression for the common factor $\gcd([5], M_5)$ provides a deeper structural understanding of the distribution of these primes, showing they form a complete translation class generated by a polynomial with coefficients in the prime field $\mathbb{F}_{19}$.

The paper explicitly states that the example was not found by brute-force search (which would be computationally infeasible due to the size of the search space, $\approx 2.2 \times 10^{15}$ candidates) but through a systematic method that reduces the problem to a finite computation. The authors note that while no counterexamples were found over prime fields in the tested range, the existence of such an example over a prime field remains an open question beyond the current computational limits.