Primes of the form $2^{\alpha}p \pm1$ with maximal rank of apparition in the Lehmer sequences

Abstract

The \emph{rank of apparition} of a prime $q$ in a given Lehmer sequence is the index of the first term in which $q$ occurs as a divisor. Furthermore, $q$ is said to have \emph{maximal rank of apparition} in an underlying Lehmer sequence provided that its rank of apparition is $q \pm 1$. Letting $p$ be a prime, in this paper we identify all primes of the form $2^{\alpha}p \pm 1$ that have maximal rank of apparition in the noted sequences.