Characterisation of Divisibility Sequences

Abstract

Certain linear recurrence sequences have a divisibility property, namely that a term un divides another term $ um$ if n divides m (e.g., the Fibonacci sequence). Such divisibility sequences can be characterised, namely they can often be written as a product of second order divisibility sequences. E.g., a power of the Fibonacci sequence will again give a divisibility sequence, but of a higher order. In this thesis we characterise divisibility sequences of orders 2, 3 and 4. A theoretical basis is provided by Ritt’s theorem on factorisation of exponential polynomials.