On certain prime numbers in Lucas sequences

In 1964, Shanks conjectured that there are infinitely many primes of the form 1/2(x(2) + 1). Therefore, the aim of this paper is to introduce a technique for studying whether or not there are infinitely many prime numbers of the form 1/2(x(2) + 1) derived from some Lucas sequences of the first kind {U-n(P, Q)}or the second kind {V-n(P, Q)}, where P >= 1 and Q = +/- 1. In addition, as applications we represent the procedure of this technique in case of x is an either integer or Lucas number of the first or the second kind with x >= 1 and 1 <= P <= 20.