Searching for Large Elite Primes

Abstract

A prime number $p$ is called elite if only finitely many Fermat numbers $2^{2^n}+1$ are quadratic residues modulo $p$. Previously, only fourteen elite primes were known explicitly, all of them smaller than $35$ million. Using computers, we searched all primes less than $10^9$ for other elite primes and discovered $p=159\,318\,017$ and $p=446\,960\,641$ as the fifteenth and sixteenth elite primes. Moreover, with another approach we found $26$ other elite primes larger than a billion, the largest of which has $1172$ decimal digits. Finally, we derive some conjectures about elite primes from the results of our computations.