Let $(F_n)_n$ be the Fibonacci sequence and $\pi(m)$ the well-known Pisano period of $m$ (i.e., the smallest period of $F_n\pmod m$).

The conjecture is that $m$ divides $\pi(m^2)$ for all $m\geq 1$ which yields the upper bound $\pi(m^2)\geq m$ (which is sharp). However, I am interested in the following much wearker inequality: $\pi(m^2)>\sqrt{m}$, for all $m\geq 1$.

Clearly, it is easy to prove that $\pi(m^2)>c\log m$ (by definition). But how to get a better lower bound for $\pi(m^2)$ (something of $O(m^{\epsilon})$, for $\epsilon\in (0,1)$)?

Someone can help me?


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