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number-theory / Function field arithmetic

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.

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number-theoryJul 14, 2026Significance 10/100Registry: site confirmed

Thakur's Conjecture on Carlitz-Wieferich Primes

Prior state unknown→disproved

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 cl…

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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.

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