On upper bounds for the count of elite primes

TL;DR

This paper investigates upper bounds on the number of elite primes related to Fermat numbers, correcting previous results and providing stronger bounds under the assumption of the Generalized Riemann Hypothesis.

Contribution

It corrects an oversight in prior work on elite primes and derives both unconditional and conditional upper bounds assuming GRH.

Findings

Corrected previous upper bound for elite primes

Derived stronger bounds assuming GRH

Identified limitations of earlier results

Abstract

We look at upper bounds for the count of certain primes related to the Fermat numbers $F_n=2^{2^n}+1$ called elite primes. We first note an oversight in a result of Krizek, Luca and Somer and give the corrected, slightly weaker upper bound. We then assume the Generalized Riemann Hypothesis for Dirichlet L functions and obtain a stronger conditional upper bound.