Primes modulo which almost all Fermat numbers are primitive roots

doi:10.1285/i15900932v30n1p133

Primes modulo which almost all Fermat numbers are primitive roots

Amin Witno

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.

DOI Code: 10.1285/i15900932v30n1p133

Keywords:
elite primes; Fermat numbers

Classification: 11A07, 11A41

Full Text: PDF

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Primes modulo which almost all Fermat numbers are primitive roots

Abstract