Abstract: The American Journal of Mathematics was America's first mathematics journal. It was founded by J.J. Sylvester in 1878 at Johns Hopkins U. It was planned that each volume would contain 384 pages in 4 parts. In the first volume Sylvester himself wrote 69 pages in three parts on binary forms (and applications to the theory of atomic structure), while Edouard Lucas wrote 88 pages in two parts (in French) on linear recurrence sequences.
Lucas' articles placed Fibonacci numbers and other ``well-known'' linear recurrence sequences into a much broader context. He examined their behaviour locally as well as globally, and he 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, and in volume 58 (in 1936) studied and tried to classify third order linear divisibility sequences; that is, linear recurrences like the Fibonacci numbers which have the property that $F_m$ divides $F_n$ whenever $m$ divides $n$. Because of many special cases, Hall was unable even to 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 will present the speaker's ongoing work in classifying all linear divisibility sequences, the key new input coming from an application of the Schmidt/Schlickewei subspace theorem from the theory of diophantine approximation.