Primes modulo which almost all Fermat numbers are primitive roots

Abstract

A prime $p$ is called elite, or anti-elite, when all but finitely many      Fermat numbers are quadratic nonresidues or residues, respectively, modulo $p$. It is known that if the multiplicative order of 2 modulo $p$ is of the form $2^s\times 5$, where $s\geq 2$, then the prime $p$ is either elite or anti-elite. Modulo elite primes of this kind, we describe some criteria by which all sufficiently large Fermat numbers be primitive roots, or all nonprimitive roots.