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A counterexample to a subadditivity conjecture of Cohen for Sophie Germain cyclic numbers

T0 review · 0 major / 3 minor · reviewed 2026-07-14 · grok-4.5

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Pith's one-line read A short length-31 window holds more Sophie Germain cyclic numbers than the initial segment [1,31], breaking Cohen's subadditivity conjecture.

desk verdict Clean, machine-checked counterexample that kills Cohen's Conjecture 66 with an explicit short window denser than [1,31]. read the letter →

arxiv 2607.09793 v1 pith:SJRQON3S submitted 2026-07-09 math.NT math.CO

Josué Alexander Ibarra This is my paper · ORCID
classification math.NTmath.CO MSC 11A2511N2511Y55
keywords SophieGermaincyclicnumberssubadditivitycounterexampleEulertotientcountingfunctionLeanformalization
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Cohen conjectured that the counting function for Sophie Germain cyclic numbers is subadditive: the count up to m+n never exceeds the sum of the separate counts up to m and up to n. The paper shows this fails already for the small pair m=31, n=3928. The reason is elementary density comparison: the open interval (3928,3959] contains eleven Sophie Germain cyclic integers while [1,31] contains only ten, so the cumulative count jumps by eleven rather than by at most ten. Because the counterexample sits well inside the range Cohen reported searching, the note also records that the earlier exhaustive search contained a coding error. The arithmetic verification is short enough to be machine-checked by the Lean 4 kernel, making the refutation both human-readable and formally certified.

What carries the argument

Sophie Germain cyclic numbers: positive integers n for which both n and 2n+1 satisfy gcd(k,φ(k))=1. Their counting function C_σ supplies the inequality that is shown to fail by a direct window comparison of length 31.

What would settle it

Recompute gcd(k,φ(k)) and gcd(2k+1,φ(2k+1)) for each of the eleven explicit values k in {3929,3931,3935,3941,3943,3945,3947,3949,3953,3957,3959}; any single failure falsifies the claimed inequality.

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Extended reading notes

Core claim

At m=31 and n=3928 one has C_σ(3959)=697 > 696 = C_σ(31)+C_σ(3928). Equivalently, the eleven integers 3929,3931,3935,3941,3943,3945,3947,3949,3953,3957,3959 are all Sophie Germain cyclic, one more than the ten such integers that lie in [1,31]. This single density excess falsifies Conjecture 66.

Load-bearing premise

That each of the eleven listed integers in the window (3928,3959] really is Sophie Germain cyclic; if any one fails the gcd condition the excess count disappears and the counterexample collapses.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 3 minor

Summary. The paper disproves Cohen's Conjecture 66 asserting subadditivity of the counting function C_σ of Sophie Germain cyclic numbers (n such that both n and 2n+1 satisfy gcd(k,φ(k))=1). It exhibits the explicit pair m=31, n=3928 for which C_σ(3959)=697>696=C_σ(31)+C_σ(3928). The short proof lists the ten Sophie Germain cyclic integers in [1,31] and the eleven in the window (3928,3959], verifies the defining conditions (with an illustrative calculation for 3929), and concludes that the denser window of length 31 violates the claimed inequality. The identical statement is formalized in Lean 4 over mathlib; the kernel accepts the proof of the negation of the conjecture using only the three standard axioms propext, Classical.choice and Quot.sound.

Significance. The note settles a conjecture that Cohen left open after an unsuccessful search to 10^6 (later attributed by him to a coding error). The counterexample is elementary, fully explicit and lies well inside the searched range, so the result is definitive rather than asymptotic. Explicit strengths include the hand-checkable lists, independent double computation of the relevant C_σ values (cross-checked against Cohen's tabulated C_σ(598)=120), and especially the machine-checked Lean formalization that never relies on native_decide. While the mathematical novelty is that of a single finite counterexample to an analog of a believed-false prime conjecture, the combination of transparency and kernel-level verification makes the note a clean, high-rigor contribution to the literature on cyclic numbers.

minor comments (3)
  1. [Theorem 1 and §2] The absolute values C_σ(3959)=697 and C_σ(3928)=686 are stated in the abstract and Theorem 1, yet the proof (and the Lean argument) only ever uses the difference of 11 versus 10. A single sentence noting the independently computed value of C_σ(3928) would make the numerical claims fully transparent without lengthening the argument.
  2. [Proof of Theorem 1] Beyond the worked example for 3929, the remaining ten window members are simply declared Sophie Germain cyclic. A compact one-line table (or footnote) recording φ(n), φ(2n+1) and the two gcds for each would render the note completely self-contained for a reader who does not consult the Lean source.
  3. [§3] The statement that #print axioms returns only the three standard axioms is valuable; a brief pointer to the precise location of the supplementary Lean file (or an inline excerpt of the key Nat.count lemmas) would make the formal claim easier to inspect.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: finite explicit enumeration of Sophie Germain cyclic numbers, machine-checked in Lean, independent of the conjecture being refuted.

full rationale

The paper's sole load-bearing claim is Theorem 1: C_σ(3959) = C_σ(3928) + 11 > C_σ(31) + C_σ(3928) because the open interval (3928, 3959] contains the eleven explicitly listed Sophie Germain cyclic integers 3929, 3931, 3935, 3941, 3943, 3945, 3947, 3949, 3953, 3957, 3959 while [1, 31] contains only the ten listed numbers 1, 2, 3, 5, 7, 11, 15, 17, 23, 29. Both lists are obtained by direct application of the classical definition gcd(n, φ(n)) = 1 (and the same for 2n + 1). No parameter is fitted, no uniqueness theorem is imported, no ansatz is smuggled via citation, and the conjecture itself is never used as a premise. The argument is further certified by a Lean 4 kernel proof that depends only on the three standard axioms propext, Classical.choice and Quot.sound; the cumulative count C_σ(3928) is never evaluated. The derivation is therefore a self-contained finite enumeration and exhibits no circular reduction.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim is a finite statement about specific integers under the classical definition of cyclic numbers. It rests solely on Euler's totient, the gcd condition, and direct enumeration of two short lists; no free parameters or newly postulated entities are required.

assumptions (3)
  • standard math n is cyclic if and only if gcd(n, φ(n))=1 (equivalently every group of order n is cyclic)
    Classical theorem used to define the objects counted by C_σ; invoked from the first paragraph onward.
  • standard math Euler's totient φ and integer gcd are correctly evaluated on the listed integers up to 7859
    Standard arithmetic; realized in Lean via mathlib Nat.totient and Nat.Coprime.
  • standard math The complete set of Sophie Germain cyclic integers in [1,31] is exactly {1,2,3,5,7,11,15,17,23,29}
    Finite exhaustive check giving C_σ(31)=10; stated in the Proof of Theorem 1.

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Cite this review

Pith. "Pith review of A counterexample to a subadditivity conjecture of Cohen for Sophie Germain cyclic numbers." pith.science (2026). https://pith.science/paper/SJRQON3S

@misc{pith2026260709793,
  author       = {Pith},
  title        = {Pith review of: A counterexample to a subadditivity conjecture of Cohen for Sophie Germain cyclic numbers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SJRQON3S}},
  note         = {Machine review of arXiv:2607.09793}
}
abstract

An integer $n \ge 1$ is cyclic if $\gcd(n,\varphi(n))=1$ (equivalently, if every group of order $n$ is cyclic), and Sophie Germain cyclic if both $n$ and $2n+1$ are cyclic. Let $C_\sigma(N)$ count the Sophie Germain cyclic integers in $[1,N]$. Cohen conjectured that $C_\sigma$ is subadditive, $C_\sigma(m+n) \le C_\sigma(m)+C_\sigma(n)$ for all $1 \le m \le n$ (his Conjecture 66), having checked $m,n \le 10^6$ without finding a counterexample. We give one: at $m=31$, $n=3928$, $C_\sigma(3959)=697 > 696 = C_\sigma(31)+C_\sigma(3928)$. The argument is short, and is verified by the Lean 4 kernel.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

4 extracted references · 1 linked inside Pith

  1. [1]

    J. E. Cohen,Conjectures about primes and cyclic numbers, J. Integer Seq.28(2025), Article 25.4.7; arXiv:2508.08335

    arXiv 2025

  2. [2]

    G. H. Hardy and J. E. Littlewood,Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes, Acta Math.44(1923), 1–70

    1923

  3. [3]

    Hensley and I

    D. Hensley and I. Richards,Primes in intervals, Acta Arith.25(1974), 375–391

    1974

  4. [4]

    Email address:alex@elninja.com

    The mathlib Community,The Lean mathematical library,https://github.com/ leanprover-community/mathlib4(2020). Email address:alex@elninja.com

    2020

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