A Pépin-Type Characterization for Fermat Pseudoprimes

Pépin’s primality test asserts that, for $n\ge 2$, $$3^{(F_n-1)\mathrm{/}2}\equiv -1(\mathrm{m}\mathrm{o}\mathrm{d}{F}_n)$$ 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{$n\ge 5$,} $$3^{(F_n-1)\mathrm{/}2}\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}{F}_n)$$ if and only if $F_n$ is pseudoprime to the base $3$.