Fermat Numbers: Pseudoprimality and Primality Constraints

TL;DR

This paper explores conditions for pseudoprimality and primality of Fermat numbers, providing new criteria based on congruences and characterizations related to Pépins test.

Contribution

It introduces necessary and sufficient conditions for Fermat number primality and pseudoprimality, expanding understanding of primality tests and their bases.

Findings

Established a necessary condition for pseudoprimality of Fermat numbers.

Derived a sufficient condition for Fermat number primality.

Characterized pseudoprimality to base 3 and other Pépins-admissible bases.

Abstract

We establish a necessary condition for pseudoprimality and a sufficient condition for primality of Fermat numbers, based on a congruence involving the exponent $(F_n-1)/4$. Moreover, in connection with P\'epin's primality test, we obtain a characterization of pseudoprimality to the base $3$ (and, more generally, to other P\'epin-admissible bases).