Prime Power Divisibility,Periodicity and Other Properties of Some Second Order Recurrences

TL;DR

This paper explores properties of second order linear recurrences, extending Wall's 1960 results on Fibonacci sequences to more general cases, with new prime power divisibility results, identities, and period derivations.

Contribution

It introduces novel prime power divisibility results and new methods for deriving periods of second order recurrences beyond Wall's original work.

Findings

New prime power divisibility results for second order recurrences

Derivation of periods using innovative methods

Establishment of new identities and matrix derivations

Abstract

Wall published a paper in 1960 on the Fibonacci sequence where he derived many results concerning the period and prime power divisibility modulo m. His periodicity results have been generalized to second order linear recurrences. Here we study the sequences generated by such recurrences, with starting values of {0,1}: among other things, we derive new prime power divisibility results, derive the period by new methods, establish new identities, show derivations involving powers of matrices generated by these general recurrences, etc.