Some algorithms for prime testing using generalized Lehmer functions

Abstract

Let N be an odd integer thought to be prime. The properties of special functions which are generalizations of the functions of Lehmer (Ann. of Math., v. 31, 1930, pp. 419-448) are used to develop algorithms that produce information concerning the possible prime divisors of N. It is shown how the factors of N ± 1 , N 2 + 1 , N 2 ± N + 1 N \pm 1,{N^2} + 1,{N^2} \pm N + 1 , together with the factor bounds on these numbers, may all be used to calculate lower bounds for the possible prime divisors of N. Frequently, these bounds are large enough that N may be shown to be prime. These tests were implemented on an IBM/370-158 computer and run on the pseudoprime divisors of the first 385 Fibonacci and Lucas numbers.