Sophie Germain Primes and the Totient of Fibonacci Numbers

Abstract

We study the set $S(q)$ of residue classes $r$ modulo the Pisano period $π(q)$ for which $q \mid φ(F_m)$ for every $m \equiv r \pmod{π(q)}$. We prove that if $q$ is a Sophie Germain prime and $z(2q+1) \mid π(q)$, then $S(q)$ is a nonempty arithmetic progression, and for $q > 5$ its cardinality is odd and $q \equiv 8 \pmod{15}$. Conversely, we show that if a prime $p \equiv 1 \pmod{q}$ has $z(p) \mid π(q)$, then necessarily $p = 2q+1$, so $q$ is Sophie Germain. We conjecture that $S(q) \neq \emptyset$ forces the existence of such a prime $p$; this is verified for all $q \leq 50000$. Assuming that $z(2q+1) \mid π(q)$ 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 $q \equiv 8 \pmod{15}$ with $(2q+1) \mid F_{π(q)}$ -- a purely Fibonacci-theoretic condition. These results generalize to arbitrary Lucas sequences $U_n(P,Q)$ with non-square discriminant.