The rank conjecture for the p-adic valuation of the Fibonacci sequence

Let $F_n$ denote the $n$th Fibonacci number, let $u_p$ be the $p$-adic valuation, and let $\alpha(p)$ be the least positive integer $n$ such that $p$ divides $F_n$. For a sequence, its rank means the rank of its $p$-kernel module. Rank conjecture. If $p\neq 2,5$ is a \prime such that $\nu_p(F_{\alpha(p)})=1$, then the rank of the sequence $\{\nu_p(F_{n+1})\}_{n\geq 0}$ is $\alpha(p)+1$. The preceding theorem establishes that this sequence is $p$-regular; the conjecture predicts the exact rank under the stated valuation condition, based on extensive computer calculations.