Primes in Sequences Associated to Polynomials (after Lehmer)

In a paper of 1933, D.H. Lehmer continued Pierce’s study of integral sequences associated to polynomials, generalizing the Mersenne sequence. He developed divisibility criteria, and suggested that prime apparition in these sequences – or in closely related sequences – would be denser if the polynomials were close to cyclotomic, using a natural measure of closeness. We review briefly some of the main developments since Lehmer’s paper, and report on further computational work on these sequences. In particular, we use Mossinghoff’s collection of polynomials with smallest known measure to assemble evidence for the distribution of primes in these sequences predicted by standard heuristic arguments. The calculations lend weight to standard conjectures about Mersenne primes, and the use of polynomials with small measure permits much larger numbers of primes to be generated than in the Mersenne case.