On the Andrews congruence for the Fibonacci quotient

TL;DR

This paper demonstrates that Andrews' 1969 Fibonacci quotient congruence implies Williams' 1991 simpler congruence, revealing a direct logical connection between two significant results in Fibonacci number theory.

Contribution

It establishes a direct implication from Andrews' older congruence to Williams' newer, simpler congruence, clarifying their relationship.

Findings

Andrews' congruence implies Williams' congruence

Simplification of Fibonacci quotient congruences

Connection between two historical results

Abstract

We show that a congruence discovered by George E. Andrews in 1969 for the Fibonacci quotient directly implies a simpler congruence found by Hugh C. Williams in 1991.