The Generalized Lucas Primes in the Landau’s and Shanks’ Conjectures

Abstract

Landau’s conjecture and Shanks’ conjecture state that there are infinitely many prime numbers of the forms x²+1 and x⁴+1 for some nonzero integer , respectively. In this paper, we present a technique for studying whether or not there are infinitely many prime numbers of the form x²+1 or x⁴+1 derived from some Lucas sequences of the first kind {U_n(P,Q)} (or simply, {U_n}) or the second kind {V_n(P,Q)} (or simply, {V_n}) , where P greater or equal to 1 and Q= 1 or -1. Furthermore, as applications we represent the procedure of this technique in case of x is either an integer or a Lucas number of the first or the second kind with x greater or equal to 1 and 1 less or equal to P less or equal to 20.