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    Flatters, Anthony 2009. Primitive divisors of some Lehmer–Pierce sequences. Journal of Number Theory, Vol. 129, Issue. 1, p. 209.

  • LMS Journal of Computation and Mathematics, Volume 3
  • January 2000, pp. 125-139

Primes in Sequences Associated to Polynomials (After Lehmer)

  • Manfred Einsiedler (a1), Graham Everest (a2) and Thomas Ward (a3)
  • DOI:
  • Published online: 01 February 2010

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.

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1.D. W. Boyd , ‘Reciprocal polynomials having small measure’. Math. Comp. 35 (1980) 13611377.

2.D. W. Boyd , ‘Speculations concerning the range of Mahler's measure’, Canad. Math. Bull. 24 (1981)453469.

3.D. W. Boyd , ‘Reciprocal polynomials having small measure II’, Math. Comp. 53 (1989)353357, S1–S5.

4.D. W. Boyd , ‘Mahler's measure and special values of L-functions’, Experiment. Math. 7 (1998) 3782.

7.C. Deninger , ‘Deligne periods of mixed motives, K-theory and the entropy of certain -actions’, J. Amer. Math. Soc. 10 (1997) 259281.

8.G. Everest and T. Ward , Heights of polynomials and entropy in algebraic dynamics (Springer, London, 1999).

11.Edmund Landau , Handbuch der Lehre von der Verteilung der Primzahlen. 2 Bände, 2nd edn, with an appendix by Paul T. Bateman (Chelsea Publishing Co., New York, 1953).

12.D. H. Lehmer , ‘Factorization of certain cyclotomic functions’, Ann. of Math. 34 (1933)461–79.

14.D. A. Lind , K. Schmidt and T. Ward , ‘Mahler measure and entropy for commuting automorphisms of compact groups’, Invent. Math. 101 (1990) 593629.

16.M. J. Mossinghoff , ‘Polynomials with small Mahler measure’,Math. Comp. 67 (1998) 16971705.

17.M. J. Mossinghoff , C. G. Pinner and J. D. Vaaler , ‘Perturbing polynomials with all their roots on the unit circle’, Math. Comp. 67 (1998) 17071726.

19.T. A. Pierce , ‘Numerical factors of the arithmetic forms Πni=1(1±αmi)’, Ann of Math. 18 (1917) 5364.

21.C. J. Smyth , ‘On the product of conjugates outside the unit circle of an algebraic integer’, Bull. London Math. Soc. 3 (1971) 169175.

22.S. S. Wagstaff , ‘Divisors of Mersenne numbers’, Math. Comp. 40 (1983) 385397.

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LMS Journal of Computation and Mathematics
  • ISSN: -
  • EISSN: 1461-1570
  • URL: /core/journals/lms-journal-of-computation-and-mathematics
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