Sophie Germain Primes and the Totient of Fibonacci Numbers

TL;DR

This paper explores the properties of Sophie Germain primes in relation to Fibonacci numbers and their totients, establishing new conditions and conjectures connecting prime types, Fibonacci periods, and Lucas sequences.

Contribution

It introduces novel results linking Sophie Germain primes to Fibonacci and Lucas sequence properties, including conditions for residue classes and prime existence implications.

Findings

If q is a Sophie Germain prime with certain divisibility conditions, then S(q) is a nonempty arithmetic progression.

For q > 5, the cardinality of S(q) is odd and q ≡ 8 mod 15.

Assuming infinitely many Sophie Germain primes satisfy a divisibility condition, the conjecture implies infinitely many primes q ≡ 8 mod 15 with (2q+1) dividing F_{π(q)}.

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 $z(2q+1) \mid \pi(q)$, 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 $z(p) \mid \pi(q)$, 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 $q \leq 50000$. 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 $q \equiv 8 \pmod{15}$ 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.