The Fibonacci sequence modulo p² -- An investigation by computer for p < 10¹⁴

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 $10^{14}$.