arXiv:2606.27885v1

Large common values of generalized Ankeny-Brauer-Chowla recurrences

Armand Noubissie, Robert F. Tichy

math.NT11D6111B3711J86

Paper metadata

Original submission: June 26, 2026
Claimed MSC: 11D61, 11B37, 11J86
arXiv categories: math.NT
Paper license: arXiv non-exclusive distribution license ↗
Comments: 15 pages

Abstract

In this paper we count the number of common values shared by two linear recurrence sequences, whose characteristic polynomials are a generalized Ankeny-Brauer-Chowla polynomial and its reciprocal. More precisely, we show that these sequences have at most two sufficiently large common values. Our proof combines Baker's theory of linear forms in logarithms of algebraic numbers with techniques from function field theory and from Galois theory.

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Audit summary

Audited against arXiv v1

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 15, 2026

01Statements2 reported findingsContains wrong statements

The Baker–determinant argument supports the intended theorem for canonically oriented cross-sequence pairs of a nonzero recurrence. Theorem 1 as printed is false: it permits the identically zero recurrence, which has infinitely many large solutions, and its ordered symmetric formulation is broader than the pairs counted in Section 4.

Intended restricted theoremCorrect

At most two canonical cross-sequence pairs under the repaired scope

Pages 2 and 9–14 · Theorem 1 and Section 4 · arXiv:2606.27885v1

Under the standing assumption that at least one initial value is nonzero, dominance makes $|A_n(u)|$ and $|A_{-m}(u)|$ separately strictly increasing beyond a uniform index. Matveev's theorem bounds every canonical cross-sequence solution by $n,m\ll(\log u)^2\log\log u$ . For three such pairs, the determinant argument splits into a nonzero-determinant case, where exponential decay contradicts that bound, and a zero-determinant case, where the second-root asymptotic forces a gap inequality incompatible with large $u$ . Thus no more than two canonically oriented cross-sequence pairs remain under the repaired scope.

Theorem 1Incorrect

The printed scope has a degenerate counterexample and the counting convention does not match the proof

Page 2 · Definition 1, Equation (3), and Theorem 1 · arXiv:2606.27885v1

The hypotheses permit $A_0=\cdots=A_{q-1}=0$ . Then $A_n(u)=0$ for every $n$ , so for every $u>c_2$ there are infinitely many distinct pairs $(n,m)$ with $n\ne m$ and $\min\{|n|,|m|\}\geq c_1$ satisfying Equation (3). Thus "at most 2 large solutions" is false as written. Independently, Equation (3) counts ordered symmetric pairs, whereas Section 4 canonically treats only collisions $|A_n(u)|=|A_{-m}(u)|$ with $n,m>0$ ; each such collision generally gives both ordered pairs in Equation (3). Repair classification: Verified scope repair. Assume at least one initial value is nonzero and state or count canonically oriented cross-sequence pairs, or explicitly quotient the original pairs by swapping.

02Proofs2 reported findingsContains incorrect or incomplete proofs

Section 4 proves at most two canonically oriented cross-sequence pairs for a nonzero recurrence. It does not prove Theorem 1 in its broader printed scope, which permits the zero recurrence and counts ordered symmetric solutions.

Section 4Correct and complete

Baker bound and three-solution determinant argument under the repaired scope

Pages 9–14 · Lemmas 7–8 and Proposition 2 · arXiv:2606.27885v1

For nonzero initial data and canonically oriented cross-sequence pairs, the dominant-root estimates make the logarithmic form nonzero and give an exponentially small upper bound; Matveev's lower bound yields $n,m\ll(\log u)^2\log\log u$ . With three such pairs, Cramer's rule either produces an integer determinant whose lower bound contradicts the exponential decay, or an exact affine dependence whose ratio of successive errors is $\gg u^{n_2-n_1}$ , again impossible for sufficiently large $u$ . The thresholds can be chosen uniformly from the displayed polynomial-degree gaps.

Proof of Theorem 1Incomplete as written

Section 4 does not cover the printed theorem's scope

Page 6 · opening of Section 3; pages 9–14 · Section 4 · arXiv:2606.27885v1

The proof uses the standing nonzero-initial-data assumption introduced only in Section 3 and selects canonically oriented cross-sign pairs. It neither treats the all-zero tuple allowed by Theorem 1 nor transfers the canonical count to the ordered symmetric solutions of Equation (3). Repair classification: Verified scope repair. Add the nonzero hypothesis and adopt the canonical counting convention stated under Part 1.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2606.27885v1
Authors listed: Armand Noubissie, Robert F. Tichy
Audit date: August 15, 2026

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If the list is empty, the section may say only “No non-novelty findings.” PART 4 — TECHNICAL AND EVIDENTIARY RULES 1. Criticism must never be overstated. Call a statement incorrect, a proof incorrect or incomplete, or a result non-new only when strong formal evidence supports that exact judgment. Otherwise use Not able to verify where that outcome exists, or remain silent in the novelty section. 2. Review a specified paper version. Give exact page, theorem, equation, or proof locations; restate relevant quantifiers and hypotheses; and check domains, endpoints, parameter ranges, constants, and dependencies explicitly. Apply a defect only to the regimes and downstream results that actually depend on it. 3. Immediate-consequence closure: when deciding whether a conclusion is supported, treat the argument as including every immediate formal consequence of facts established earlier in the paper and available at that point. An immediate formal consequence is a local, mechanically checkable inference using only definitions already in force, elementary logic, direct substitution or renaming, weakening of an inequality or set inclusion, or a monotonicity or symmetry property explicit in a definition or already proved. 4. Such an inference must introduce no new lemma, estimate, construction, case analysis, limiting or compactness argument, external result, or new issue of uniformity or parameter dependence. Check the direction of monotonicity and the dependence of every threshold or constant explicitly. Do not exchange pointwise and uniform conclusions, change exceptional sets, interchange quantifiers or limits, or extend an endpoint under this rule unless the required fact was already established. 5. If the claimed conclusion follows after immediate-consequence closure, it is fully supported. 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P(delta)

is proved for every sufficiently small

delta>0

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P

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P(delta)

holds for every

delta>0

requires no correction; the omitted monotonicity step is exposition rather than an audit finding. 6. Immediate-consequence closure permits reconstruction of an omitted inference; it does not permit silent alteration of the printed domain, hypotheses, or quantifiers of an intermediate statement later invoked outside its stated scope. If formal validity requires changing the text of that statement or its stated range, report the appropriate finding even when the repair is easy. Thus a result stated only for

t>0

and subsequently applied at

t=0

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Large common values of generalized Ankeny-Brauer-Chowla recurrences

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

At most two canonical cross-sequence pairs under the repaired scope

The printed scope has a degenerate counterexample and the counting convention does not match the proof

Baker bound and three-solution determinant argument under the repaired scope

Section 4 does not cover the printed theorem's scope

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