In mathematics, a divisibility sequence is an integer sequence such that for all natural numbers m, n,
i.e., 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 natural numbers m, n,
Note that a strong divisibility sequence is immediately a divisibility sequence; if , immediately . Then by the strong divisibility property, and therefore .
- 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 F = (1, 1, 2, 3, 5, 8,...) form a strong divisibility sequence.
- More generally, Lucas sequences of the first kind are divisibility sequences.
- Elliptic divisibility sequences are another class of such sequences.
- 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