Let $a_n$ denote the Fibonacci numbers, for a prime $p$ let $\alpha(p)$ denote the first index $n$ such that $p|a_n$ and let $r$ denote the golden ratio.

Q: Is there a proof of $\lim_{x\rightarrow \infty}\frac{1}{x^2}\sum_{\alpha(p)\leq x} \log p = \frac{3 \log r}{\pi^2}$ ?

The sum is taken over all primes $p$ that divide at least one Fibonacci number with index less or equal to $x$.

It is related to the number of Wall-Sun-Sun primes (cf. e.g. Is the Crandall, Dilcher and Pomerance heuristic concerning Wall-Sun-Sun primes still state of the art?).

Added later: It is known (P. Kiss, Primitive Divisors of Lucas Numbers, "Application of Fibonacci Numbers" (A. N. Phillipou et al.,Ed.), pp. 29 -- 38, Kluwer Acad. Publ. 1988) that $\limsup_{x\rightarrow \infty}\frac{1}{x^2}\sum_{\alpha(p)\leq x} \log p \leq \frac{3 \log r}{\pi^2}$.

2more comments