The Fibonacci sequence modulo $p^2$ -- An investigation by computer for $p < 10^{14}$

TL;DR

This paper investigates the period lengths of Fibonacci sequences modulo prime squares for primes less than 10^14, revealing that the period length modulo p^2 is never equal to that modulo p, and establishing a general formula for these periods.

Contribution

The paper provides extensive computational evidence that the period length modulo p^2 differs from that modulo p for large primes and proves a general formula for the period lengths of Fibonacci sequences modulo prime powers.

Findings

Period length modulo p^2 is never equal to that modulo p for primes less than 10^14.

Established the formula (p^n) = (p)  p^{n-1} for all primes less than 10^14.

Extensive computational search supports theoretical results.

Abstract

We show that for primes $p < 10^{14}$ the period length $\kappa (p^2)$ of the Fibonacci sequence modulo $p^2$ is never equal to its period length modulo $p$. The investigation involves an extensive search by computer. As an application, we establish the general formula $\kappa (p^n) = \kappa (p) \cdot p^{n-1}$ for all primes less than $10^{14}$.