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

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 $10^{14}$.