Testing numbers of the form N = 2kp(m) - 1 for primality

We suggest an algorithm to test numbers of the form N = 2kp(m) - 1 for primality, where 2k < p(m), k is an odd positive integer, 2k < p(m), p is a prime number, and p = 3 (mod 4). The algorithm makes use of the Lucas functions. First we present an algorithm to test numbers of the form N = 2k3(m) - 1. Then the same technique is used in the more general case where N = 2kp(m) - 1. The algorithms suggested here are of complexity O((log N)(2) log log N log log log N).