Number Theory Web Seminar: Andrew Granville (Université de Montréal)

Linear Divisibility sequences

In 1878, in the first volume of the first mathematics journal published in the US, Edouard Lucas wrote 88 pages (in French) on linear recurrence sequences, placing Fibonacci numbers and other linear recurrence sequences into a broader context. He examined their behaviour locally as well as globally, and asked several questions that influenced much research in the century and a half to come.

In a sequence of papers in the 1930s, Marshall Hall further developed several of Lucas' themes, including studying and trying to classify third order linear divisibility sequences; that is, linear recurrences like the Fibonacci numbers which have the additional property that $F_m$ divides $F_n$ whenever $m$ divides $n$. Because of many special cases, Hall was unable to even conjecture what a general theorem should look like, and despite developments over the years by various authors, such as Lehmer, Morgan Ward, van der Poorten, Bezivin, Petho, Richard Guy, Hugh Williams,... with higher order linear divisibility sequences, even the formulation of the classification has remained mysterious.

In this talk we present our ongoing efforts to classify all linear divisibility sequences, the key new input coming from a wonderful application of the Schmidt/Schlickewei subspace theorem from the theory of diophantine approximation, due to Corvaja and Zannier.

For further information about the seminar, please visit this webpage.

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