A Pépin-Type Characterization for Fermat Pseudoprimes

Abstract

Pépin's primality test asserts that, for $n\ge 2$, \[ 3^{(F_n-1)/2}\equiv -1 \pmod{F_n} \] if and only if $F_n$ is prime. We establish a natural analogue of Pépin's criterion for pseudoprimality. More precisely, we prove that, for \mbox{$n\ge 5$,} \[ 3^{(F_n-1)/2}\equiv 1 \pmod{F_n} \] if and only if $F_n$ is pseudoprime to the base $3$.

arXiv commentSubstantially revised version.5 pages