Fermat Numbers: Pseudoprimality and Primality Constraints

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).