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We suggest an algorithm to test numbers of the form  N  = 2 kp m   − 1 for primality, where 2 k  &lt;  p m  ,  k  is an odd positive integer, 2 k  &lt;  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  = 2 k 3  m   − 1. Then the same technique is used in the more general case where  N  = 2 kp   m   − 1. The algorithms suggested here are of complexity  O ((log  N ) 2  log log  N  log log log  N ).

Testing numbers of the form N = 2 kp m − 1 for primality