Some primality tests that eluded Lucas

Abstract

In his extensive memoir on the sequences that now bear his name, Lucas provided some primality tests for numbers $$N$$N, where $$N\pm 1$$N±1 is divisible by a large prime power. No proofs were provided for these tests, and they are not correct as stated. Nevertheless, it is possible to correct these tests and make them more general. The purpose of this paper is to make these corrections and then show how the ideas behind these tests can be extended to numbers $$N$$N, where $$N^2+1$$N2+1 is divisible by a large prime power. In order to do this we develop further the properties of a certain pair of sequences which satisfy a linear recurrence relation of order 4.