bibkey: jaffe2026denominators
authors: Grace Natalie Jaffe
year: 2026
title: "Denominators in the Farey Sequence, the Stern Sequence, and the Frobenius Problem with Relatively Prime Multipliers"
doi: null
url: https://arxiv.org/abs/2610.17267v1
claim: "Conjecture 7.3 bounds g(a,c(a,b))/g(a,b) by 5 for every coprime pair of integers a,b at least 3, where c(a,b) is the least integer greater than b and coprime to a, and g uses primitive nonnegative coefficients."
strata_touched:
  - D5/S3/ArithUnits/JaffeSuccessorRatioRefutation
license: CC-BY-4.0
triage: anchor

Primitive nonnegative representations and the successor-ratio conjecture

Verified locator

The versioned source is arXiv:2610.17267v1. Its primary HTML, Section 3, defines [ R(a,b)={ax+by:x,y\in\mathbb Z_{\geq0},\ \gcd(x,y)=1}. ] The quantity (g(a,b)) is the greatest integer in (\mathbb Z\setminus R(a,b)), with infinity when there is no greatest missing integer. The axes are included: the coefficients are nonnegative, as stated in Section 3. The introduction’s positive-coefficient wording does not replace this definition.

Section 7, Conjecture 7.3, states:

For all a,b >= 3 relatively prime and for c(a,b) the minimum integer satisfying both c(a,b)>b and gcd(c(a,b),a)=1, we have g(a,c(a,b))/g(a,b) <= 5.

There is no global (a<b) convention. The local ordered hypotheses in Section 2 and the ordered dataset following Conjecture 7.3 are separate conditions. Section 7 also considers fixed (b=3) and larger (a). Symmetry of (g) does not make its asymmetric successor operation invariant under exchange of the two arguments.

The retained primary HTML has 451,273 bytes and SHA-256 3ea624b4ac5a553f9b0b348a7f9079d096f20871ced150f1a593b1022cc66f9f. The retained text has 47,585 bytes and SHA-256 71e5ef50ec26c31589d69dee838a778f6ccdad99eb90dc09356505ed526d95ac. The HTML identifies the source license as CC BY 4.0. No external proof code is transplanted.

Earlier and related statements

The same author’s Topics in analytic number theory, University of Illinois Urbana-Champaign dissertation, 2023, states this same assertion as Conjecture 8.3, printed page 43 / PDF page 51. The institutional record is https://www.ideals.illinois.edu/items/127306 and the persistent handle is https://hdl.handle.net/2142/120085. The checked PDF has 1,895,118 bytes and SHA-256 2f668879c4ec48a6b292a79bb35cbf617ad3c1d2a08ebd7d2c038fa512e36259. Its relevant conjecture, numerical discussion and appendix do not supply a whole proof or refutation. The thesis and paper name one mathematical target.

Shunpei Shimizu’s A factor-four lower bound for the Frobenius problem with relatively prime multipliers (Jaffe’s Conjecture 7.2), draft v0.1.1, DOI https://doi.org/10.5281/zenodo.23480212, concerns the different factor-four lower bound. The inspected main manuscript does not settle the successor-ratio Conjecture 7.3. It is not used as a theorem here.

Refutation and prior ingredients

The formal source is D5/S3/ArithUnits/JaffeSuccessorRatioRefutation.lean. Its Represented predicate uses nonnegative integer coefficients and their actual gcd; sourceG selects the greatest missing integer and returns infinity when one does not exist. fullClaim quantifies over all coprime integers (a,b\geq3), with neither coordinate order assumed. The designated theorem result has the literal type Not fullClaim.

The counterexample is [ a=87841397512641630,\qquad b=7,\qquad c(a,7)=49. ] Every integer from 8 through 48 shares a factor with (a), whereas (\gcd(a,7)=\gcd(a,49)=1). Scaling a missing integer for ((a,7)) by 7 gives a missing integer for ((a,49)): any purported primitive representation would force the first coefficient to be divisible by 7, and division would preserve primitive coefficients. Consequently the actual maxima satisfy (g(a,49)\geq7g(a,7)>0), so their well-defined quotient exceeds 5.

Finiteness is proved, rather than assumed. If (H(n)) counts all nonnegative solutions and (P(n)) the primitive ones, a gcd partition and Möbius inversion give [ ABP(n)\geq\varphi(n)-AB,\tau(n), \qquad n\tau(n)^2\leq16^7\varphi(n)^2, ] where (\tau(n)) is the number of positive divisors of (n). Thus every (n>16^7(AB)^2) is represented. The missing integer 1 and this universal tail give a positive greatest missing integer in (\mathbb Z). The maxima themselves are not evaluated or replaced by prefix maxima.

The source’s Theorem 4.3 already supplies the finiteness ingredient in ordinary mathematics, and Theorem 7.1 supplies the scaling ingredient. Their mathematical content is attributed to the source. The Lean proof checks these ingredients using finite counting, divisor factorization and coprimality; it does not assume the paper’s theorems as axioms. The coarse divisor-count cutoff is sufficient for the comparison and is not an optimal bound.

Qualification boundary

Registration #20244 anchors the whole Conjecture 7.3. The bounded qualification checked the versioned source, the relevant thesis passages, the related 7.2 manuscript, and arXiv, OpenAlex, Crossref and targeted web results. No inspected text gave a whole answer to this exact target. Citation indexing was incomplete: the checked 7.2 manuscript cites the source although its citation-index query returned zero records. Unindexed work, alternate terminology and unvisited sources remain outside the checked scope. No global priority or exhaustive absence claim follows.

The prior bounded repository, pinned Mathlib and third-party Lean searches identified Möbius inversion, totient factorization and integer extrema APIs, but no exact primitive-tail or whole successor-ratio result. Ordinary FrobeniusNumber APIs omit coefficient coprimality and do not close this gap. The pinned Mathlib revision is 56a005855ea17e0b0958f73ec801ae140b0a1679 with Lean 4.35.0. This is one unfrozen candidate for one target; independent mirror review, freezing, integration and accepted credit are separate obligations. The ordered variant with (a<b) remains unresolved here.