The exact gcd conjecture for the degree-five Carlitz–Wieferich factor

Let $q=19^3$, let

$$\mu(X)=X^5+5X^3+3X^2-4X-9\in\mathbf{F}_{19}[X],\qquad G=\mu(T^q-T)\in\mathbf{F}_q[T],$$

and let $[5]=T^{q^5}-T$ and $M_5$ be Thakur's polynomial. The polynomial $P\in\mathbf{F}_{19^3}[T]$ is the degree-five $c$-Wieferich prime exhibited in the paper, and its $6859=q$ translates are $P(T-a)$ for $a\in\mathbf{F}_{19^3}$. Exact gcd conjecture.

$$\gcd\bigl([5],M_5\bigr)=G$$

in $\mathbf{F}_{19^3}[T]$. Equivalently, the $c$-Wieferich primes of degree $5$ in $\mathbf{F}_{19^3}[T]$ are exactly the $6859$ translates $P(T-a)$, with $a\in\mathbf{F}_{19^3}$. The theorem preceding this conjecture proves that $G$ divides both $[5]$ and $M_5$ and is the product of these translates; the conjecture asserts that no further common factor exists.