Sophie Germain Primes and the Totient of Fibonacci Numbers
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914498744418304 |
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| author | Goel, Aradhya |
| author_facet | Goel, Aradhya |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_17847 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sophie Germain Primes and the Totient of Fibonacci Numbers Goel, Aradhya Number Theory 11B39 (Primary) 11A25, 11A41 (Secondary) 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. |
| title | Sophie Germain Primes and the Totient of Fibonacci Numbers |
| topic | Number Theory 11B39 (Primary) 11A25, 11A41 (Secondary) |
| url | https://arxiv.org/abs/2604.17847 |