Sophie Germain Primes and the Totient of Fibonacci Numbers

We study the set $S(q)$ of residue classes $r$ modulo the Pisano period $\pi (q)$ for which $q\mid \phi (F_m)$ for every $m\equiv r(\mathrm{m}\mathrm{o}\mathrm{d}\pi (q))$. We prove that if $q$ is a Sophie Germain prime and $z(2q+1)\mid \pi (q)$, then $S(q)$ is a nonempty arithmetic progression, and for $q>5$ its cardinality is odd and $q\equiv 8(\mathrm{m}\mathrm{o}\mathrm{d}15)$. Conversely, we show that if a prime $p\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}q)$ has $z(p)\mid \pi (q)$, then necessarily $p=2q+1$, so $q$ is Sophie Germain. We conjecture that $S(q)\mathrm{\ne }\mathrm{\varnothing }$ forces the existence of such a prime $p$; this is verified for all $q\le 50000$. Assuming that $z(2q+1)\mid \pi (q)$ holds for infinitely many Sophie Germain primes (verified computationally for approximately 23.9% of them), the Sophie Germain conjecture implies the existence of infinitely many primes $q\equiv 8(\mathrm{m}\mathrm{o}\mathrm{d}15)$ with $(2q+1)\mid F_{\pi (q)}$ – a purely Fibonacci-theoretic condition. These results generalize to arbitrary Lucas sequences $U_n(P,Q)$ with non-square discriminant.