Divisibility sequence

In mathematics, a divisibility sequence is an integer sequence ${\displaystyle (a_{n})}$ indexed by positive integers n such that

${\displaystyle {\text{if }m\mid n{\text{ then }a_{m}\mid a_{n}$

for all m, n. 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 ${\displaystyle (a_{n})}$ such that for all positive integers m, n,

${\displaystyle \gcd(a_{m},a_{n})=a_{\gcd(m,n)}.}$

Every strong divisibility sequence is a divisibility sequence: ${\displaystyle \gcd(m,n)=m}$ if and only if ${\displaystyle m\mid n}$. Therefore, by the strong divisibility property, ${\displaystyle \gcd(a_{m},a_{n})=a_{m}$ and therefore ${\displaystyle a_{m}\mid a_{n}$.

Examples

• Any constant sequence is a strong divisibility sequence.
• Every sequence of the form ${\displaystyle a_{n}=kn,}$ for some nonzero integer k, is a divisibility sequence.
• The numbers of the form ${\displaystyle 2^{n}-1}$ (Mersenne numbers) form a strong divisibility sequence.
• The repunit numbers in any base R_n^(b) form a strong divisibility sequence.
• More generally, any sequence of the form ${\displaystyle a_{n}=A^{n}-B^{n}$ for integers ${\displaystyle A>B>0}$ is a divisibility sequence. In fact, if ${\displaystyle A}$ and ${\displaystyle B}$ are coprime, then this is a strong divisibility sequence.
• The Fibonacci numbers F_n form a strong divisibility sequence.
• More generally, any Lucas sequence of the first kind U_n(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

• 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 (3): 577–584. doi:10.2307/2370976. 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. doi:10.2307/2374733. 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 (eds.), Number Theory, Analysis and Geometry. In Memory of Serge Lang, Springer, pp. 243–271, ISBN 978-1-4614-1259-5