divisibility sequence

English

Noun

divisibility sequence (plural divisibility sequences)

  1. (number theory, algebra) Any sequence of integers {an}, indexed by the natural numbers, such that if n is divisible by m then an is divisible by am.
    • 2012, P. Ingram, J. H. Silverman, “Primitive Divisors in Elliptic Divisibility Sequences”, in Dorian Goldfeld, Jay Jorgenson, Peter Jones, Dinakar Ramakrishnan, Kenneth Ribet, John Tate, editors, Number Theory, Analysis and Geometry, Springer,, page 244:
      If 𝒞=(Cn)n≥1 is any divisibility sequence, one says that a prime p is a primitive divisor of Cn if p|Cn but p∤C1C2Cn−1. Primitive divisors of certain divisibility sequences were studied by Zsigmondy [37] in the 19th century.
    • 2013, Graham Everest, Thomas Ward, Heights of Polynomials and Entropy in Algebraic Dynamics, Springer, page 138:
      These divisibility sequences satisfy the same recurrence relations as the polynomials ϕn and ψn2 (see Appendix C).
    • 2021, Masum Billal, Samin Riasat, Integer Sequences, Springer, page 59:
      Moreover, in order to keep a divisibility sequence normalized, we can assume without loss of generality that a0 = 0 and a1=1.

Usage notes

The concept can be generalised to sequences of elements of any ring for which divisibility is defined.

Derived terms

  • strong divisibility sequence

Translations

Further reading

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