On the solutions of the equation p=x^2+y^2+1 in Lucas sequences

Abstract

In 1970, Motohashi proved that there are an infinite number of primes having the form p=x^2+y^2+1 for some nonzero integers x and y. In this paper, we present a technique for studying the solutions of the equation p=x^2+y^2+1, where the unknowns are derived from some Lucas sequences of the first kind {Un(P, Q)} or the second kind {Vn(P, Q)} with P and Q are certain nonzero relatively primes integers. As applications to this technique, we apply our procedure in case of (x, y, p) = (Ui(P, Q), Uj(P, Q), Uk(P, Q)) or (Vi(P, Q), Vj(P, Q), Vk(P, Q))  with i, j, k > 1, -2 < p < 3 and Q =1 or -1.