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Let $(F_n)$ the Fibonacci sequence and $\pi(m)$ the Pisano period of $m$ (i.e., the smallest period of $F_n \pmod{m}$). There are many proved results about $\pi(m)$. For example, it is known that $\pi(p^a)\mid p^{a-1}\pi(p)$, for any prime number $p$ and integer $a\geq 1$. It is conjectured that $\pi(p^a)= p^{a-1}\pi(p)$.

My question is a weak version of this conjecture. Is it possible at least to prove that $\pi(m^2)>\sqrt{m}$ (for all $m$) or something like that?

Thanks in advance!

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    $\begingroup$ You'd better know Lucas' repetition law. See, e.g., Dickson's book Histroy of Number Theory.. $\endgroup$
    – Zhi-Wei Sun
    Commented May 24, 2019 at 16:51
  • $\begingroup$ @Zhi-WeiSun I know this rescult, but I didn't understand the connection. Could you please explain me better? Is it true that $p^a\mid \pi(p^{2a})$? $\endgroup$
    – Pierre
    Commented May 24, 2019 at 21:54
  • $\begingroup$ For what it's worth, Pisano periods are tabulated out to 10,000 at oeis.org/A001175/b001175.txt $\endgroup$
    – Gerry Myerson
    Commented May 24, 2019 at 23:50
  • $\begingroup$ @Zhi-WeiSun Also, this result you mentioned is a very easy consequence of a Lengyel formula for p-adic valuation of F_n. I don't find connection with my problem. $\endgroup$
    – Pierre
    Commented May 25, 2019 at 17:26
  • $\begingroup$ @Zhi-WeiSun Is it possible to prove at least that $\pi(m^2)>\sqrt{m}$ or something like that? $\endgroup$
    – Pierre
    Commented May 27, 2019 at 2:16

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