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arxiv: 2604.17847 · v3 · submitted 2026-04-20 · 🧮 math.NT

Sophie Germain Primes and the Totient of Fibonacci Numbers

Aradhya Goel (Indian Institute of Technology , Kanpur) This is my paper

Pith reviewed 2026-05-10 04:10 UTC · model grok-4.3

classification 🧮 math.NT
keywords Sophie Germain primesPisano periodFibonacci totientrank of appearanceLucas sequencesarithmetic progressionsentry point
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The pith

If q is a Sophie Germain prime and z(2q+1) divides π(q), then S(q) is a nonempty arithmetic progression of odd cardinality with q ≡ 8 mod 15 for q > 5.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines S(q) as the set of residue classes r modulo the Pisano period π(q) such that the prime q divides the Euler totient of the Fibonacci number F_m for every index m congruent to r. It proves that when q is Sophie Germain and satisfies the extra condition that the rank of appearance z(2q+1) divides π(q), this set S(q) must be a nonempty arithmetic progression. For primes q larger than 5 the length of the progression is odd and q lies in the residue class 8 modulo 15. A converse statement shows that any prime p congruent to 1 modulo q whose rank of appearance divides π(q) is forced to equal 2q+1. The same conclusions hold more generally for Lucas sequences with nonsquare discriminant.

Core claim

We prove that if q is a Sophie Germain prime and z(2q+1) | π(q), then S(q) is a nonempty arithmetic progression, and for q > 5 its cardinality is odd and q ≡ 8 mod 15. Conversely, we show that if a prime p ≡ 1 mod q has z(p) | π(q), then necessarily p = 2q+1, so q is Sophie Germain. We conjecture that S(q) ≠ ∅ forces the existence of such a prime p; this is verified for all q ≤ 50000. Assuming that z(2q+1) | π(q) holds for infinitely many Sophie Germain primes, the Sophie Germain conjecture implies the existence of infinitely many primes q ≡ 8 mod 15 with (2q+1) | F_π(q).

What carries the argument

The set S(q) of residue classes r modulo the Pisano period π(q) for which q divides φ(F_m) for all m ≡ r mod π(q), together with the rank of appearance z(n) of an integer n in the Fibonacci sequence.

If this is right

  • S(q) is a nonempty arithmetic progression whenever q is Sophie Germain and z(2q+1) divides π(q).
  • For q > 5 the cardinality of S(q) is odd and q ≡ 8 mod 15.
  • Any prime p ≡ 1 mod q with z(p) | π(q) must equal 2q+1.
  • The same arithmetic-progression structure holds for the analogous sets defined by arbitrary Lucas sequences U_n(P,Q) with nonsquare discriminant.
  • Under the assumption that z(2q+1) | π(q) for infinitely many Sophie Germain primes, the Sophie Germain conjecture yields infinitely many q ≡ 8 mod 15 satisfying the pure Fibonacci condition (2q+1) | F_π(q).

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The verified conjecture that nonempty S(q) forces existence of p = 2q+1 offers a potential computational route to search for Sophie Germain primes by examining Fibonacci totient patterns instead of direct primality tests on 2q+1.
  • If the divisibility z(2q+1) | π(q) holds for a positive-density subset of Sophie Germain primes, the results would translate the Sophie Germain conjecture into an equivalent statement purely about divisibility in the Fibonacci sequence at the Pisano period.
  • The same machinery may apply to other linear recurrence sequences, potentially linking additional families of primes to arithmetic-progression structures in their associated periods.

Load-bearing premise

That z(2q+1) divides π(q) whenever q is a Sophie Germain prime.

What would settle it

A Sophie Germain prime q larger than 5 for which z(2q+1) does not divide π(q) yet S(q) remains nonempty, or a prime q ≤ 50000 with nonempty S(q) but no prime p ≡ 1 mod q satisfying z(p) | π(q).

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper defines the set S(q) of residue classes r modulo the Pisano period π(q) such that q divides φ(F_m) for every m ≡ r mod π(q). It proves that if q is a Sophie Germain prime and z(2q+1) | π(q), then S(q) is a nonempty arithmetic progression; for q > 5 the cardinality is odd and q ≡ 8 mod 15. Conversely, any prime p ≡ 1 mod q with z(p) | π(q) must equal 2q+1. The authors conjecture that S(q) ≠ ∅ implies existence of such a p (verified computationally for all q ≤ 50000) and, assuming z(2q+1) | π(q) for infinitely many SG primes (verified for ~23.9%), deduce that the Sophie Germain conjecture implies infinitely many q ≡ 8 mod 15 with (2q+1) | F_π(q). The results generalize to Lucas sequences U_n(P,Q) with nonsquare discriminant.

Significance. If the conditional results hold, the work establishes a concrete link between Sophie Germain primes, the entrypoint function z, Pisano periods, and the totient of Fibonacci numbers. The proven implication and converse are rigorous, the computational support for the conjecture is substantial, and the generalization to Lucas sequences is a clear strength. The conditional infinitude statement, while dependent on an open divisibility, offers a new Fibonacci-theoretic formulation that could be of interest to researchers studying prime divisors of Fibonacci numbers.

major comments (2)
  1. [Abstract] Abstract and main theorem statement: the structural claim that S(q) is a nonempty arithmetic progression with odd cardinality (q > 5) and q ≡ 8 mod 15 is conditional on the unproven divisibility z(2q+1) | π(q); this holds for only ~23.9% of SG primes by computation and is not established in general, so the result remains conditional on an assumption whose infinitude is also open.
  2. [Abstract] Converse and conjecture paragraph: while the converse (any qualifying p must be 2q+1) is proven, the conjecture that S(q) ≠ ∅ forces existence of such a p is supported solely by verification up to q = 50000; no theoretical argument is supplied, leaving the implication from nonempty S(q) to the existence of the Sophie Germain prime 2q+1 open.
minor comments (2)
  1. [Abstract] The generalization to arbitrary Lucas sequences is announced but the precise adaptations of the proofs (especially the arithmetic-progression structure and the mod-15 congruence) are not sketched; a brief outline would improve readability.
  2. Notation: the definition of S(q) would benefit from an explicit small-q example (e.g., q=5 or q=11) showing the residue classes explicitly.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their careful reading, positive evaluation of the significance, and recommendation for minor revision. We address each major comment below and will incorporate clarifications into the abstract.

read point-by-point responses

  1. Referee: [Abstract] Abstract and main theorem statement: the structural claim that S(q) is a nonempty arithmetic progression with odd cardinality (q > 5) and q ≡ 8 mod 15 is conditional on the unproven divisibility z(2q+1) | π(q); this holds for only ~23.9% of SG primes by computation and is not established in general, so the result remains conditional on an assumption whose infinitude is also open.

    Authors: We agree that the structural result on S(q) being a nonempty arithmetic progression (with the stated cardinality and congruence properties for q > 5) is conditional on the divisibility z(2q+1) | π(q). This hypothesis is already stated explicitly both in the abstract and in the main theorem. Our computations confirm the divisibility for approximately 23.9% of Sophie Germain primes, but we make no general claim that it holds for all such primes or for infinitely many. The subsequent conditional deduction (that the Sophie Germain conjecture implies infinitely many q ≡ 8 mod 15 with (2q+1) | F_π(q)) assumes both the Sophie Germain conjecture and the infinitude of the divisibility. We will revise the abstract to add an explicit remark noting that the divisibility condition is not known to hold infinitely often. revision: partial

  2. Referee: [Abstract] Converse and conjecture paragraph: while the converse (any qualifying p must be 2q+1) is proven, the conjecture that S(q) ≠ ∅ forces existence of such a p is supported solely by verification up to q = 50000; no theoretical argument is supplied, leaving the implication from nonempty S(q) to the existence of the Sophie Germain prime 2q+1 open.

    Authors: The converse—that any prime p ≡ 1 mod q with z(p) | π(q) must equal 2q+1—is proven rigorously in the paper. The further claim that S(q) ≠ ∅ implies the existence of such a prime p is presented as a conjecture, backed by exhaustive computational verification for all q ≤ 50000. We do not supply a theoretical proof of this implication, as none is currently available, and we acknowledge that the implication remains open. revision: no

standing simulated objections not resolved
  • The conjecture that S(q) ≠ ∅ implies existence of the prime p = 2q+1 lacks a theoretical proof and rests only on computational verification.

Circularity Check

0 steps flagged

No circularity; conditional theorems rely on explicit external assumptions and standard number-theoretic definitions

full rationale

The paper proves its main structural claims about S(q) directly from the definitions of the Pisano period π(q), the rank of appearance z(p), and the totient function φ, under the explicitly stated hypothesis that z(2q+1) divides π(q) for Sophie Germain prime q. The converse direction is shown by a direct argument that any prime p ≡ 1 mod q satisfying the divisibility condition must equal 2q+1. The conjecture that S(q) nonempty implies existence of such a p is presented separately and supported only by finite verification up to q ≤ 50000; it is not used inside the proofs. No self-citations, fitted parameters renamed as predictions, or ansatzes smuggled via prior work appear in the derivation chain. All steps remain independent of the target conclusions.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The paper relies entirely on established mathematical concepts and definitions from number theory without introducing new free parameters, ad hoc axioms, or invented entities.

axioms (3)
  • standard math The Fibonacci sequence satisfies the standard recurrence F_n = F_{n-1} + F_{n-2} with F_1=1, F_2=1.
    This is the foundational definition used to define periods and divisibility.
  • standard math Pisano period π(q) exists for any q and is the period of the Fibonacci sequence modulo q.
    This is a well-known property in number theory.
  • standard math Properties of the entrypoint function z(n) and Sophie Germain primes.
    Standard definitions in the field of number theory.

pith-pipeline@v0.9.0 · 5566 in / 1870 out tokens · 59803 ms · 2026-05-10T04:10:07.287818+00:00 · methodology

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Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

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    T. Koshy,Fibonacci and Lucas Numbers with Applications, Wiley-Interscience, New York, 2001. 11 Table 1: Sophie Germain primesq≤5000 withS(q)̸=∅. q p= 2q+ 1π(q)z(p)π(q)/z(p) 5 p 3 7 8 8 1−1 5 11 20 10 2 +1 23 47 48 16 3−1 53 107 108 36 3−1 83 167 168 168 1−1 173 347 348 116 3−1 293 587 588 588 1−1 443 887 888 888 1−1 593 1187 1188 1188 1−1 653 1307 1308 43...

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