arXiv:2509.09578v2

Repdigits as Product of Consecutive Shifted Tribonacci Numbers

Pranabesh Das, Salah Eddine Rihane, Alain Togbé

math.NT11B3911J86

Paper metadata

Original submission: September 11, 2025
Latest revision: July 1, 2026 · version 2
Claimed MSC: 11B39, 11J86
arXiv categories: math.NT
Paper license: arXiv non-exclusive distribution license ↗
Comments: Accepted

Abstract

A repdigit is a positive integer that has only one distinct digit in its decimal expansion, i.e., a number has the form $d(10^m-1)/9$ for some $m\geq 1$ and $1 \leq d \leq 9$ . Let $\left(T_n\right)_{n\ge0}$ be the Tribonacci sequence. This paper deals with the presence of repdigits in the products of consecutive shifted Tribonacci numbers.

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Audited against arXiv v2

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 findingsCorrect

The four nonexistence results for repdigits among products of consecutive shifted Tribonacci numbers are correct. The printed asymptotic error exponents used in their proofs are too strong, but replacing them by the verified $O(\alpha^{-n})$ bounds leaves ample room in the explicit constants and the reduction argument.

Theorems 1.1–1.4Correct

The four repdigit classifications survive the corrected error estimates

Pages 2 and 4–12 · Theorems 1.1–1.4 and Sections 3–5 · arXiv:2509.09578v2

Writing each shifted factor as $c_\alpha\alpha^{n+i}+u_i$ gives $|u_i|\leq1+\alpha^{-n/2}$ . Expanding a product of at most four factors therefore gives relative error $O(\alpha^{-n})$ , rather than the printed $O(\alpha^{-3n/2})$ . With this correction, Matveev's theorem still gives initial bounds below $5.8\times10^{16}$ for $n$ in the three proof patterns. The convergents displayed in the paper satisfy the de Weger separation inequalities for the corrected logarithmic offsets and these larger bounds, and the reported finite searches cover the remaining ranges. Thus the same reductions prove all four conclusions.

Theorems 1.3 and 1.4Typo

The variable $k$ is omitted from the quantified tuple

Page 2 · statements of Theorems 1.3 and 1.4 · arXiv:2509.09578v2

Both statements display products involving $T_{n+k}$ but quantify only the positive integers $n,\ell,m,d$ . The quantified list must also include $k$ . The equations, the conditions $k>\ell$ or $k\geq\ell$ , and the proofs all use $k$ as an independent positive integer, so the correction is unique and does not change the asserted result.

02Proofs3 reported findingsContains incorrect or incomplete proofs

The Diophantine-reduction strategy is valid after a verified repair, but equations (3.4), (4.3), and (5.3) state false error bounds. The logarithmic offsets in the de Weger steps also contain mechanically identifiable notation errors.

Equations (3.4), (4.3), and (5.3)Incorrect as written

The product errors decay like $\alpha^{-n}$ , not $\alpha^{-3n/2}$

Pages 5, 7, and 9 · equations (3.4), (4.3), and (5.3) · arXiv:2509.09578v2

Every factor is shifted by $\pm1$ . Relative to its dominant Binet term, that shift alone has order $\alpha^{-n}$ , so a uniform $O(\alpha^{-3n/2})$ estimate is false. Repair classification: Verified repair. Replace the three exponents by $-n$ . A direct product estimate keeps the numerical right-hand constants $11964$ , $3988$ , and $8721506$ valid. Reapplying Matveev gives $n<3.59\times10^{16}$ , $n<3.57\times10^{16}$ , and $n<5.73\times10^{16}$ in the three cases. The paper's subsequent convergents still meet the required separation thresholds with these corrected bounds, so the reduction and finite verification close unchanged.

de Weger reductionsTypo

The displayed logarithmic offsets omit powers and normalization

Pages 6, 8, and 10–11 · applications of Lemma 2.3 · arXiv:2509.09578v2

After division by $\log 10$ , the offset must contain $\log(d/(9c_\alpha^{\ell}))/\log10$ in the first two patterns and $\log(d/(9c_\alpha^{k+\ell}))/\log10$ in the mixed pattern. The printed definitions omit the exponent and the division by $\log10$ . These corrections are forced by the linear forms immediately above them. Substituting the corrected offsets into the displayed convergents gives positive distances larger than the corresponding de Weger thresholds, so the numerical exclusions remain valid.

Sections 3–5Correct and complete

Linear forms, reduction, and finite ranges

Pages 4–12 · proofs of Theorems 1.1–1.4 · arXiv:2509.09578v2

After the verified error repair above, the nonzero linear forms satisfy the hypotheses of Matveev's theorem and provide explicit initial bounds. The continued-fraction reduction eliminates every index above the listed computational cutoffs. The remaining bounded cases are searched directly, and the symmetry obtained by changing the two shifts supplies the compressed proof of Theorem 1.4 without introducing a new estimate or case.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2509.09578v2
Authors listed: Pranabesh Das, Salah Eddine Rihane, Alain Togbé
Audit date: August 15, 2026

01Establish identityMatch an authenticated scholarly identity to this paper.
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You are a mathematical auditor. Evaluate only the correctness of the paper's central statements, the correctness and completeness of its proofs, and evidence-backed non-novelty. Do not make editorial comments or evaluate importance, exposition, style, or suitability for publication. Keep the report neutral, precise, and professional. PART 1 — CORRECTNESS OF THE MAIN STATEMENTS Identify the paper's central statements and assess each one separately using exactly one of these outcomes: - Correct: the statement has a correct, verifiable proof; or, despite a defect in the printed proof, you can give a fully verified repair or independent proof. - Incorrect: you can give strong formal evidence that the statement as written is false, preferably a counterexample or a complete contradiction. - Not able to verify: the supplied proof does not establish the statement, and you can neither supply a fully verified proof nor disprove it. 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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. 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P(delta)

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P

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P(delta)

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delta>0

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t>0

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Repdigits as Product of Consecutive Shifted Tribonacci Numbers

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The four repdigit classifications survive the corrected error estimates

The variable kkk is omitted from the quantified tuple

The product errors decay like α−n\alpha^{-n}α−n, not α−3n/2\alpha^{-3n/2}α−3n/2

The displayed logarithmic offsets omit powers and normalization

Linear forms, reduction, and finite ranges

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Authenticate with ORCID, then require an exact arXiv match

The variable $k$ is omitted from the quantified tuple

The product errors decay like $\alpha^{-n}$ , not $\alpha^{-3n/2}$