Probable Prime Tests Using Lucas Sequences

Abstract

In [1] Baillie & Wagstaff proposed that a strong probable prime test combined with a strong Lucas probable prime test would be a swift and powerful probable prime test. Even today, 16 years after it was proposed, no number has been found for which the test fails. This test for the primality of n includes a search for a quadratic non-residue D mod n, and in the worst case this search may take many steps $(O({n^{\frac{1}{4} + \in }})with \in >0)$ with ε > 0). It will be shown how to avoid this search in 7 out of 8 cases of n mod 24 by explicitly producing quadratic non-residues.