Divisibility sequence

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In mathematics, a divisibility sequence is an integer sequence indexed by positive integers n such that

for all mn. That is, whenever one index is a multiple of another one, then the corresponding term also is a multiple of the other term. The concept can be generalized to sequences with values in any ring where the concept of divisibility is defined.

A strong divisibility sequence is an integer sequence such that for all positive integers mn,

Every strong divisibility sequence is a divisibility sequence: if then . Then by the strong divisibility property, and therefore .

Examples[edit]

  • Any constant sequence is a strong divisibility sequence.
  • Every sequence of the form , for some nonzero integer k, is a divisibility sequence.
  • Every sequence of the form for integers is a divisibility sequence.
  • The Fibonacci numbers Fn form a strong divisibility sequence.
  • More generally, any Lucas sequence of the first kind Un(P,Q) is a divisibility sequence. Moreover, it is a strong divisibility sequence when gcd(P,Q)=1.
  • Elliptic divisibility sequences are another class of such sequences.

References[edit]

  • Everest, Graham; van der Poorten, Alf; Shparlinski, Igor; Ward, Thomas (2003). Recurrence Sequences. American Mathematical Society. ISBN 978-0-8218-3387-2.
  • Hall, Marshall (1936). "Divisibility sequences of third order". Am. J. Math. 58: 577–584. JSTOR 2370976.
  • Ward, Morgan (1939). "A note on divisibility sequences". Bull. Amer. Math. Soc. 45 (4): 334–336. doi:10.1090/s0002-9904-1939-06980-2.
  • Hoggatt, Jr., V. E.; Long, C. T. (1973). "Divisibility properties of generalized Fibonacci polynomials" (PDF). Fibonacci Quarterly: 113.
  • Bézivin, J.-P.; Pethö, A.; van der Porten, A. J. (1990). "A full characterization of divisibility sequences". Am. J. Math. 112 (6): 985–1001. JSTOR 2374733.
  • P. Ingram; J. H. Silverman (2012), "Primitive divisors in elliptic divisibility sequences", in Dorian Goldfeld; Jay Jorgenson; Peter Jones; Dinakar Ramakrishnan; Kenneth A. Ribet; John Tate, Number Theory, Analysis and Geometry. In Memory of Serge Lang, Springer, pp. 243–271, ISBN 978-1-4614-1259-5