arXiv:2607.20763v1

On a Diophantine Equation with Jacobsthal and Fibonacci Numbers

Daeyeoul Kim, Zekiye Pinar Cihan, Zeynep Demirkol Ozkaya, Ilker Inam

math.NT11D4511J86

Paper metadata

Original submission: July 22, 2026
Claimed MSC: 11D45, 11J86
arXiv categories: math.NT
Paper license: CC BY 4.0 ↗
Comments: 12 pages. Final version. To appear in Hacettepe Journal of Mathematics and Statistics
DOI: 10.15672/hujms.1823402 ↗

Abstract

In the present paper, we identify all Jacobsthal numbers that may be expressed as a product of three Fibonacci numbers. More precisely, our main result shows that the only solution to the Diophantine equation $F_kF_lF_m=J_n$ for $2<k<l<m$ is $(k,l,m,n)=(5,7,8,12).$ The proof relies on techniques involving linear forms in logarithms.

AI-generated audit

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

01Statements1 reported findingContains unsupported statements

No counterexample was found to the claimed unique solution. Theorem 2.1 is not able to be verified because two parameter-dependent continued-fraction reductions are applied using single numerical values that fail for other admissible parameters. Exact enumeration confirms the theorem inside the final claimed box $m\leq236$ , but the printed argument does not validly reduce every solution to that box.

Theorem 2.1Not able to verify

The asserted complete classification is not established

Pages 2 and 9–11 · Theorem 2.1 and final reduction · arXiv:2607.20763v1

The exact claim is that $F_kF_lF_m=J_n$ with $2<k<l<m$ has only $(k,l,m,n)=(5,7,8,12)$ . The linear-form estimates give a large preliminary bound, and an exact-integer enumeration verifies the stated solution is unique whenever $m\leq236$ . However, the two Dujella–Pethő steps intended to prove $l\leq114$ and then $m\leq236$ do not establish positive epsilon uniformly over the still-variable indices. Concrete admissible parameter values make the displayed epsilon negative. Consequently the proof does not cover all possible solutions, and no independent proof or counterexample is supplied here.

Published version of the reviewed result ↗

02Proofs8 reported findingsContains incorrect or incomplete proofs

The second and third continued-fraction reductions contain unresolved uniformity gaps, and the final finite verification is absent. Two earlier numerical calculations are also incorrect as written, although local repairs preserve the preliminary bounds. Four mechanically determined notation defects are reported separately as typos.

Second and third Dujella–Pethő reductionsIncomplete as written

The quoted epsilon values are not uniform in the remaining indices

Pages 10–11 · Reductions following Equations (4.16) and (4.21) · arXiv:2607.20763v1

For the second reduction, $\mu_k=\log(5/(3F_k))/\log\alpha$ varies with $k$ , but the paper uses one convergent $q_{92}$ and the single value $\varepsilon=0.00120395$ to treat every $k\leq110$ . With the printed $q_{92}$ and $M=8.24\times10^{42}$ , direct high-precision reevaluation at the admissible value $k=98$ gives $\|\mu_{98}q_{92}\|\approx0.00513103040825$ and $M\|\tau q_{92}\|\approx0.0153446398986$ , hence $\varepsilon\approx-0.0102136094904$ . Lemma 3.3 is therefore inapplicable there. Likewise, the third reduction has $\mu_{k,l}=\log(\sqrt5/(3F_kF_l))/\log\alpha$ but uses one value for all $k\leq110$ , $l\leq114$ ; at $(k,l)=(72,109)$ its printed $q_{93}$ and $M=2.04107\times10^{43}$ give $\varepsilon\approx-0.0357465065164$ . A separate positive-epsilon convergent, or another reduction, is required for every uncovered parameter choice. Repair classification: No repair supplied.

Dujella–Pethő reduction lemma ↗

End of the proof of Theorem 2.1Incomplete as written · verified local repair

The finite box is not checked in the manuscript

Page 11 · Final two sentences before the acknowledgements · arXiv:2607.20763v1

After asserting $m\leq236$ , the proof immediately says it is finished; it never checks the finitely many triples $3\leq k<l<m\leq236$ . An exact-integer enumeration, generating $F_r$ and $J_s$ from their defining recurrences and comparing every product $F_kF_lF_m$ with the Jacobsthal values for $n<3m$ , returns only $(5,7,8,12)$ . This repairs the omitted finite step itself, but it does not repair the earlier failure to establish $m\leq236$ .

Full paper, version 1 ↗

First Dujella–Pethő reductionIncorrect as written · verified repair

The displayed epsilon omits the subtraction term

Page 9 · Computation using $q_{92}$ after Equation (4.23) · arXiv:2607.20763v1

Lemma 3.3 defines $\varepsilon=\|\mu q\|-M\|\tau q\|$ . For the printed $q_{92}$ and $M=6.13\times10^{43}$ , the quoted $0.284374$ is $\|\mu q_{92}\|$ alone; the omitted term is approximately $0.114153692450$ , so the actual epsilon is approximately $0.170220186049$ . It remains positive, and the corrected threshold is $k<110.481$ , which still yields $k\leq110$ . Repair classification: Verified repair. This local correction does not resolve the later parameter-dependent gaps.

Dujella–Pethő reduction lemma ↗

Third Matveev estimateIncorrect as written · verified local repair

The product uses a smaller height bound than the one established

Page 8 · Display immediately after the choice of $A_3''$ · arXiv:2607.20763v1

The proof establishes and chooses $A_3''=4.25\times10^{24}(1+\log(3m))^2$ , but the next application of Matveev's theorem multiplies by $4.23\times10^{24}(1+\log(3m))^2$ . The smaller value has not been justified. Retaining $4.25$ gives the valid bound $\log|\Lambda_3|>-2.808\times10^{36}(1+\log(3m))^3$ and, conservatively, $m<5.95\times10^{42}$ in place of $5.93\times10^{42}$ . Repair classification: Verified local repair. The later choices of $M$ must be adjusted accordingly, and the uniformity gap remains.

Matveev's lower-bound theorem ↗

Fibonacci characteristic polynomialTypo

The constant term has the wrong sign

Page 3 · Paragraph following Equation (4.2) · arXiv:2607.20763v1

The polynomial is printed as $\Psi(X)=X^2-X+1$ . Replace $+1$ by $-1$ , giving $\Psi(X)=X^2-X-1$ . The displayed roots $\alpha=(1+\sqrt5)/2$ and $\beta=(1-\sqrt5)/2$ , and every subsequent use of $\alpha\beta=-1$ , uniquely determine the correction. The argument itself consistently uses the corrected polynomial.

Full paper, version 1 ↗

Statement of Matveev's theoremTypo

The defining linear form is missing its final minus one

Page 2 · Theorem 3.1, Equation (3.5) · arXiv:2607.20763v1

The cited theorem prints $\Lambda=\alpha_1^{b_1}\cdots\alpha_t^{b_t}$ , but the stated lower bound and all three later applications require $\Lambda=\alpha_1^{b_1}\cdots\alpha_t^{b_t}-1$ . Insert $-1$ . The applications define $\Lambda_1,\Lambda_2,\Lambda_3$ in that corrected form, so this notation defect does not itself alter them.

Matveev's lower-bound theorem ↗

Height estimate for $3F_k/5$Typo

The Fibonacci exponent uses $l$ instead of $1$

Page 6 · Last displayed estimate · arXiv:2607.20763v1

The bound is printed with $(k-l)\log\alpha$ and is followed immediately by the bound $\log(15)+k\log\alpha$ . Replace $k-l$ by $k-1$ , using $F_k\leq\alpha^{k-1}$ from Equation (4.3). The intended correction is unique and the next line already uses a valid weaker consequence.

Full paper, version 1 ↗

Continued-fraction threshold displaysTypo

The inequality directions contradict the invoked lemma and the conclusions

Pages 9–11 · Displays immediately before $k\leq110$ , $l\leq114$ , and $m\leq236$ · arXiv:2607.20763v1

Lemma 3.3 says there is no solution when $w\geq\log(Aq/\varepsilon)/\log B$ . Therefore an existing solution must satisfy $w<\log(Aq/\varepsilon)/\log B$ . Replace the printed lower-bound signs in the three concluding displays by upper-bound signs. This is mechanically forced by the upper bounds stated on the following lines. Correcting these symbols does not repair the missing positive-epsilon checks in the second and third reductions.

Dujella–Pethő reduction lemma ↗

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2607.20763v1
Authors listed: Daeyeoul Kim, Zekiye Pinar Cihan, Zeynep Demirkol Ozkaya, Ilker Inam
Audit date: August 15, 2026

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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. For every adverse or unverified outcome, restate the exact claim and hypotheses, identify its location in the reviewed version, and give the decisive reasoning. Do not infer that a theorem is false merely because its printed proof is defective. Assign the overall statements status by the following priority: 1. Contains wrong statements, if at least one central statement is Incorrect. 2. Contains unsupported statements, if no central statement is Incorrect but at least one is Not able to verify. 3. Correct, if every central statement is Correct, apart from any typographical or minor formal findings allowed by Part 4. PART 2 — CORRECTNESS AND COMPLETENESS OF THE PROOFS Audit the proofs of the central statements and every material internal or cited input on which they depend. For each proof or proof segment, use the narrowest supported outcome: - Correct and complete: every substantive step is verifiable and no case or crucial nontrivial step is missing. - Incorrect as written: a displayed calculation, inference, lemma, or cited application is formally false or inapplicable. - Incomplete as written: a necessary substantive step or case is missing, even though no explicit contradiction has been proved. - Not able to verify: the available paper and sources are insufficient to decide whether the step is valid. Do not flag omitted routine details, compressed arguments, or “analogously” steps when the omitted reasoning is recoverable and correct. For every defect, identify its smallest exact location, explain its downstream dependency, and classify any proposed repair as one of: - Verified repair: supply enough reasoning to verify that the repair works throughout every affected use. - Plausible repair only: describe the possible repair but do not treat it as established. - No repair supplied. Assign the overall proofs status by the following priority: 1. Contains incorrect or incomplete proofs, if at least one proof is Incorrect as written or Incomplete as written. 2. Contains unverified proofs, if there is no proved incorrectness or incompleteness but at least one material proof is Not able to verify. 3. Correct, if all audited proofs are Correct and complete, apart from any typographical or minor formal findings allowed by Part 4. PART 3 — NOVELTY OF THE MAIN STATEMENTS This is a one-sided, evidence-only check. Evaluate only central results. Report a result only when a verifiable primary source, publicly available before the earliest public version of the paper that contains the result, proves the same statement up to a change of notation, or when the result follows directly and immediately from such an earlier theorem. For each non-novelty finding, provide the earlier source, its public date, the exact theorem or proposition, and an explicit hypothesis-and-conclusion mapping. A later source cannot establish non-novelty. Do not flag a result merely because an easier proof exists or because it follows through a nontrivial argument. Do not flag a result that the paper itself explicitly presents as known, a survey item, or an alternative proof. Mention only statements that meet this non-novelty threshold. Say nothing about statements for which no such evidence was found: do not certify novelty, list unsuccessful searches, or add statement-level “no prior source found” entries. 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. Do not report a gap, unsupported statement, minor formal correction, or recommendation to add an explanatory sentence, and do not mention the suppressed item in the report. This suppression does not apply to a genuine typographical error as defined below, because a typo is a literal notation defect that still requires a textual correction. For example, if

P(delta)

is proved for every sufficiently small

delta>0

and

P

is immediately monotone in the required direction, then the claim that

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

remains a formal scope mismatch unless the endpoint case was separately established before the application. 7. Apply this mandatory gate before recording any adverse, unverified, typographical, or minor formal finding: (a) identify the exact assertion or inference alleged to be defective; (b) reconstruct the strongest conclusion following from the preceding argument after all admissible immediate formal consequences are added; (c) identify the specific defect that remains — a formally demonstrated falsehood, a concrete nontrivial unresolved proof obligation, or a genuine domain, hypothesis, quantifier, or notation mismatch requiring a textual correction; and (d) state the evidence and minimal repair. If no specific defect remains, or if the only possible addition would be explanatory, suppress the finding entirely. Use Not able to verify only when a precise, nontrivial unresolved obligation can be identified. 8. A typographical error is an incorrect symbol, index, exponent, sign, variable name, or notation mismatch whose unique intended correction is mechanically determined from the surrounding text and does not alter the mathematical argument. Report it in yellow as Typo, with the printed text, correction, exact location, and reason it is harmless. It does not lower an overall status. 9. A minor formal correction is a boundary, endpoint, or displayed parameter-range defect that can be repaired consistently in every affected statement by a local change, without a new argument, without invalidating any related result or application, and without creating another substantive defect. Report it in yellow as Minor formal correction, with the same details. It does not lower an overall status. 10. Do not use either minor category if more than one materially different repair is possible; if the repair changes the substance or scope of a central result; if an excluded case is used later; if the correction fails to propagate consistently; or if a new mathematical idea is required. In such cases use the strongest substantive classification actually supported by evidence. 11. Write every mathematical expression as valid, compilable LaTeX. Use

...

for inline mathematics and

...

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On a Diophantine Equation with Jacobsthal and Fibonacci Numbers

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The asserted complete classification is not established

The quoted epsilon values are not uniform in the remaining indices

The finite box is not checked in the manuscript

The displayed epsilon omits the subtraction term

The product uses a smaller height bound than the one established

The constant term has the wrong sign

The defining linear form is missing its final minus one

The Fibonacci exponent uses lll instead of 111

The inequality directions contradict the invoked lemma and the conclusions

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The Fibonacci exponent uses $l$ instead of $1$