Sophie Germain primes and the totient of Fibonacci numbers

Abstract: We study the set $S(q)$ of residue classes $r$ modulo the Pisano period $\pi(q)$ for which $q \mid \varphi(F_m)$ for every $m \equiv r \pmod{\pi(q)}$. We prove that if $q$ is a Sophie Germain prime and $z(2q+1) \mid \pi(q)$, where $z$ denotes the rank of apparition, then $S(q)$ is a nonempty arithmetic progression; for $q > 5$, its cardinality is odd and $q \equiv 8 \pmod{15}$. Conversely, if a prime $p \equiv 1 \pmod{q}$ has $z(p) \mid \pi(q)$, 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 $q \leq 50{,}000$. Assuming the divisibility $z(2q+1) \mid \pi(q)$ holds for infinitely many Sophie Germain primes (verified for approximately $23.9\%$ of the $669$ Sophie Germain primes $q \leq 50{,}000$), the Sophie Germain conjecture implies the existence of infinitely many primes $q \equiv 8 \pmod{15}$ 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.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory seminar (CANT 2026)