divisibility sequence

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divisibility sequence

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divisibility sequence (plural divisibility sequences)

1. (number theory, algebra) Any sequence of integers {a_n}, indexed by the natural numbers, such that if n is divisible by m then a_n is divisible by a_m.
   • 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 ${\displaystyle {\mathcal {C}}=(C_{n})_{n\geq 1}}$ is any divisibility sequence, one says that a prime ${\displaystyle p}$ is a primitive divisor of ${\displaystyle C_{n}}$ if ${\displaystyle p\vert C_{n}}$ but ${\displaystyle p\nmid C_{1}C_{2}C_{n-1}}$. Primitive divisors of certain divisibility sequences were studied by Zsigmondy in the 19^th 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 ${\displaystyle \phi _{n}}$ and ${\displaystyle \psi _{n}^{2}}$ (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 ${\displaystyle a_{0}}$ = 0 and ${\displaystyle a_{1}=1}$.

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

• strong divisibility sequence

• Elliptic divisibility sequence on Wikipedia.Wikipedia
• Fibonacci number § Divisibility properties on Wikipedia.Wikipedia