Fermat-Wilson Supercongruences, arithmetic derivatives and strange factorizations

TL;DR

This paper explores supercongruences related to Fermat and Wilson congruences over finite fields, revealing new links with arithmetic derivatives, zeta values, and prime factorizations in function field arithmetic.

Contribution

It systematically expands on the first analog of supercongruences, introducing mixed derivatives and establishing new equivalences and prime factorizations.

Findings

Relations linking supercongruences with arithmetic derivatives and zeta values.

New equivalences involving mixed derivatives.

Prime factorizations involving derivative conditions for key function field quantities.

Abstract

In [Tha15], we looked at two (`multiplicative' and `Carlitz-Drinfeld additive') analogs each, for the well-known basic congruences of Fermat and Wilson, in the case of polynomials over finite fields. When we look at them modulo higher powers of primes, i.e. at `supercongruences', we find interesting relations linking them together, as well as linking them with arithmetic derivatives and zeta values. In the current work, we expand on the first analog and connections with arithmetic derivatives more systematically, giving many more equivalent conditions linking the two, now using `mixed derivatives' also. We also observe and prove remarkable prime factorizations involving derivative conditions for some fundamental quantities of the function field arithmetic.