To let $ (F_n) $ the Fibonacci sequence and $ pi (m) $ the pisano time of $ m $ (ie the smallest period of $ F_n pmod {m} $). There are many proven results over $ pi (m) $, For example, this is known $ pi (p ^ a) mid p ^ {a-1} pi (p) $, for any prime $ p $ and an integer $ a geq 1 $, it is supposed that $ pi (p ^ a) = p ^ {a-1} pi (p) $,

My question is a weak version of this assumption. For example, I want to prove that $ m ^ 2 mid pi (m) $ for all positive integers $ m $,

Since for $ m = p_1 ^ {a_1} cdots p_k ^ {a_k} $it holds $ pi (m) = lcm ( pi (p_1 ^ {a_1}), lpoints, pi (p_k ^ {a_k})) $Then it is enough to prove that

[

p^amid pi(p^{2a}),

]

For every prime $ p $ and an integer $ a geq 1 $, There's the guess $ pi (p ^ a) = p ^ {2a-1} pi (p) $We have some freedom, though $ a <2a-1 $ (ie. $ a> 1 $). Could someone help me?

Actually, even in the case of $ m mid pi (m ^ 3) $ would be helpful. Thank you in advance!