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Einsiedler, Manfred; Everest, Graham; Ward, Thomas

Primes in sequences associated to polynomials (after Lehmer). (English) Zbl 1037.11014

LMS J. Comput. Math. 3, 125-139 (2000).
Summary: In a paper of 1933, D. H. Lehmer [Ann. Math. (2) 34, 461-479 (1933; Zbl 0007.19904)] 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.

Cited in 3 Documents

MSC:

11C08 Polynomials in number theory
11A51 Factorization; primality

Citations:

Zbl 0007.19904
Cite Review PDF
Full Text: DOI Link

Online Encyclopedia of Integer Sequences:

a(n+6) = -a(n+5) + a(n+4) + 3a(n+3) + a(n+2) - a(n+1) - a(n). a(n) = sign(n) if abs(n)<=3.
The periodic point counting sequence for the toral automorphism given by the polynomial of (conjectural) smallest Mahler measure. The map is x -> Ax mod 1 for x in [0,1)^10, where A is the companion matrix for the polynomial x^10+x^9-x^7-x^6-x^5-x^4-x^3+x+1.
Number of orbits of length n under the automorphism of the 3-torus whose periodic points are counted by A001945.
Number of orbits of length n in map whose periodic points are A059928.
A divisibility sequence derived from Lehmer’s polynomial x^10+x^9-x^7-x^6-x^5-x^4-x^3+x+1. Square root of the terms in A059928.

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