Characterisation of linear recurrences whose ratio is integer infinitely often

Number theory 1 Minute

You need to know: Set {\mathbb N} of natural numbers, complex numbers, notation {\mathbb C}[x] for the set of polynomials with complex coefficients.

Background: A linear recurrence is a sequence of complex numbers \{G(n)\}_{n\in{\mathbb N}} such that G(n+k)=c_0 G(n) + \dots + c_{k-1} G(n+k-1), \, n\in{\mathbb N} for some k\geq 1 and complex numbers c_0, c_1, \dots, c_{k-1}.

The Theorem: On 12th November 2001, Pietro Corvaja and Umberto Zannier submitted to Inventiones Mathematicae a paper in which they proved that for any linear recurrences F(n) and G(n) such that F(n)/G(n) is an integer for infinitely many values of n, there exists a nonzero polynomial P(X) \in {\mathbb C}[X] and positive integers q, r such that both sequences \{P(n)F(qn+r)/G(qn+r)\}_{n\in{\mathbb N}} and \{G(qn+r)/P(n)\}_{n\in{\mathbb N}} are linear recurrences.

Short context: In 1988, van der Poorten, confirming a conjecture of Pisot,
proved that if the ratio F(n)/G(n) of linear recurrences F(n), G(n) is an integer for all large n, then F(n)/G(n) is itself a linear recurrence. In 1989, van der Poorten asked whether a similar result can be proved under much weaker assumption that F(n)/G(n) is an integer infinitely open. This is what the Theorem achieves.

Links: The original paper is available here.

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Published by Bogdan Grechuk

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