On the rank of apparition of composite N in the Lehmer sequences

Abstract

The rank of apparition of a number N in an underlying recursively defined Lehmer sequence is the index of the first term that contains N as a divisor. In this paper, we state the conjecture that if ( N ± 1 ) / 2 is the rank of apparition of N then N is prime. Evidence of its truth is supplied by demonstrating the impossibility of ( N ± 1 ) / 2 as being the rank of apparition of any composite number defined by the product of distinct odd primes. The conjecture, if true, will provide an extension to the primality result of D. Lehmer which states that if N ± 1 is the rank of apparition of N then N is prime.