arXiv:2607.25168v1

On certain $D(9)$ and $D(64)$ Diophantine triples

Benjamin Earp-Lynch, Simon Earp-Lynch, Omar Kihel

math.NT11D0911D4511B3711J86

Paper metadata

Original submission: July 28, 2026
Claimed MSC: 11D09, 11D45, 11B37, 11J86
arXiv categories: math.NT
Paper license: arXiv non-exclusive distribution license ↗
Comments: 28 pages. This is a pre-print of an article published in Acta Mathematica Hungarica. The final published version is available at https://doi.org/10.1007/s10474-020-01061-2
Journal reference: Acta Math. Hungar., 162 (2) (2020), 483-517
DOI: 10.1007/s10474-020-01061-2 ↗

Abstract

A set of $m$ distinct positive integers $\{a_{1},\dots a_{m}\}$ is called a $D(q)$ - $m$ -tuple for nonzero integer $q$ if the product of any two increased by $q$ , $a_{i}a_{j}+q$ , $i\neq j$ is a perfect square. Due to certain properties of the sequence, there are many $D(q)$ -Diophantine triples related to the Fibonacci numbers. A result of Baćić and Filipin characterizes the solutions of Pellian equations that correspond to $D(4)$ -Diophantine triples of a certain form. We generalize this result in order to characterize the solutions of Pellian equations that correspond to $D(l^2)$ -Diophantine triples satisfying particular divisibility conditions. % Subsequently, we employ this result and bounds on linear forms in logarithms of algebraic numbers in order to classify all $D(9)$ and $D(64)$ -Diophantine triples of the form $\{F_{2n+8},9F_{2n+4},F_{k}\}$ and $\{F_{2n+12},16F_{2n+6},F_{k}\}$ , where $F_{i}$ denotes the $i$ th Fibonacci number.

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

01Statements3 reported findingsContains wrong statements

The central Pellian classification in Lemma 3.1 is false as stated, with an explicit $D(64)$ counterexample. The two advertised Fibonacci-triple classifications are not disproved, but their proofs depend on that classification and do not address the omitted solution classes.

Lemma 3.1Incorrect

The claimed classification of all Pellian solutions is false

Pages 6–9 · Lemma 3.1, especially the inference after the bound $b'c'<l^2+1$ · arXiv:2607.25168v1

Take $l=8$ , $a=960$ , $b=2640$ , and $c=1$ . Then $a<b<a(4+4/l^2)$ , $l^2\mid a$ , and $ab+l^2=1592^2,\qquad ac+l^2=32^2,\qquad bc+l^2=52^2.$ Thus $(t,s)=(52,32)$ is a positive solution of $at^2-bs^2=l^2(a-b)$ arising from a $D(64)$ -triple. It is not in either class printed in Lemma 3.1. For exponent $\nu=1$ , the two printed classes give $s=1592-960=632$ and $s=1592+960=2552$ , corresponding to $c=416$ and $c=6784$ ; subsequent multiplication by the positive fundamental unit increases these positive coefficients. The proof's decisive implication is invalid: $b'c'<l^2+1$ together with $b'c'+l^2$ being a square does not force $b'c'=0$ . Repair classification: No repair supplied; additional solution classes must be classified.

Theorem 1.1Not able to verify

The $D(9)$ Fibonacci-triple classification is not established

Pages 2 and 10–19 · Theorem 1.1 and Sections 4–8 · arXiv:2607.25168v1

The reduction to the two sequences $C_j^{\pm}$ uses Lemma 3.1 to assert that those sequences exhaust all positive solutions of the Pellian equation. The proof of that exhaustion contains the false inference identified above. The later logarithmic estimates and finite reductions analyze only the two selected classes and therefore cannot exclude an omitted class. No counterexample to Theorem 1.1 was found, but no independent complete classification of the relevant $l=3$ solution classes is supplied.

Theorem 1.2Not able to verify

The $D(64)$ Fibonacci-triple classification is not established

Pages 2 and 19–28 · Theorem 1.2 and Sections 9–13 · arXiv:2607.25168v1

For $3\mid n$ , the proof invokes precisely the $l=8$ , $l^2\mid a$ branch of Lemma 3.1. The explicit counterexample to that branch shows that the asserted two-class parametrization is not valid under those hypotheses. Consequently, reducing and checking the displayed sequences $C_j^{(\pm)}$ does not exclude Fibonacci values arising from another Pellian class. Direct exact checks for the tested small parameter range support the claimed answer, but they do not repair the missing all- $n$ classification. No counterexample to Theorem 1.2 itself was found.

02Proofs4 reported findingsContains incorrect or incomplete proofs

The proof of the foundational Pellian classification contains a formally false inference and the resulting parametrizations are not exhaustive. A second Pell-equation argument omits one solution class but has a verified repair. The numerical reductions within the displayed classes were independently reproduced.

Proof of Lemma 3.1Incorrect as written

A small-product bound is incorrectly converted into vanishing

Page 8 · Paragraph following the estimate $b'c'<l^2+1$ · arXiv:2607.25168v1

From $b'c'<l^2+1$ and $b'c'+l^2=u^2$ , the proof immediately concludes $b'=0$ . That conclusion is false: a square strictly between $l^2$ and $2l^2+1$ may exist, for example $(l+1)^2=l^2+2l+1$ when $l>2$ . The concrete $l=8$ triple in Part 1 confirms that this is not merely a loose estimate but corresponds to a genuinely omitted positive solution class. This defect occurs before the descent is used to parametrize every solution, so it propagates to both main theorem proofs. Repair classification: No repair supplied.

Lemma 2.7Incomplete as written · verified repair

One Pell-equation solution class is omitted

Pages 4–5 · Proof of Lemma 2.7 · arXiv:2607.25168v1

The proof says that all solutions of $X^2-5Y^2=4$ arise from $(\pm3+\sqrt5)(9+4\sqrt5)^j$ , but it omits the class $2(9+4\sqrt5)^j$ , whose first positive member is $18+8\sqrt5$ . The omitted class does not invalidate Lemma 2.7: its $X$ -coordinates are $L_{6j}$ , and Binet's formulas give $F_{6j+1}<L_{6j}<F_{6j+2}$ for $j\geq1$ , so none is a Fibonacci number. Repair classification: Verified repair by adding this class and the displayed strict comparison. This restores the multiplicative-independence input used later, but it does not repair Lemma 3.1.

Sections 5–8 and 10–13Correct and complete within the stated classes

The reductions inside the two displayed Pellian classes check out

Pages 11–19 and 20–28 · linear-form bounds and Baker–Davenport reductions · arXiv:2607.25168v1

The height estimates, logarithmic-form inequalities, and continued-fraction reductions were checked with the paper's constants. For the $D(9)$ calculation, all $84$ pairs consisting of $1\leq n\leq42$ and either sign admit a Dujella–Pethő certificate excluding every integer $j\geq7$ ; the largest resulting lower cutoff is approximately $6.62049$ . For the $D(64)$ calculation, all $86$ pairs with $1\leq n\leq43$ give a cutoff below $6$ , the largest being approximately $5.85150$ . Exact recurrence arithmetic then confirms the paper's finite checks. These verifications do not establish exhaustiveness because that earlier step depends on Lemma 3.1.

Definition of $U_j,V_j$ in Section 9Typo

The factor $1/2$ is missing from the defining unit

Page 20 · Definition immediately following the Pellian parametrization · arXiv:2607.25168v1

The displayed definition uses $(F_{2n+9}+\sqrt{F_{2n+6}F_{2n+12}})^j$ . Replace the base by $((F_{2n+9}+\sqrt{F_{2n+6}F_{2n+12}})/2)^j$ . The preceding parametrization, the subsequent definition of $\beta_n$ , and every later formula all contain this factor. The correction is unique and does not affect the intended calculations.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

Detailed audit reportFull reasoning, manuscript locations, and references.

Open report PDF ↗

Author response

Challenge an audit finding

Local workflow preview

A listed author may submit formal evidence that an audit is inaccurate. The response would be considered in a fresh AI re-evaluation; it would not edit the audit automatically.

Paper: arXiv:2607.25168v1
Authors listed: Benjamin Earp-Lynch, Simon Earp-Lynch, Omar Kihel
Audit date: August 15, 2026

01Establish identityMatch an authenticated scholarly identity to this paper.
02Submit evidenceIdentify the finding and give a formal mathematical response.
03Re-evaluateA separate agent checks the response and records a disposition.

Email fallback for papers without a linked ORCID

A production fallback could send a one-time link only when the submitted address matches an independently maintained author-contact allowlist for this paper. MathAudit must return the same message for every address so the form cannot reveal which contacts are on that list.

Structured response preview

This form remains unavailable until production identity verification succeeds. Nothing entered here is submitted.

Audit finding challengedWhy the evaluation is inaccurateFormal evidence or repairVerifiable sources, if anyI am submitting evidence, not instructions for the reviewing agent.

This panel never establishes authorship in the local prototype. A production result should be described narrowly as an authenticated ORCID match or control of a separately allowlisted author-contact mailbox.

Audit controls

Status: Report ready
Source: arXiv v1
Prompt: Standard prompt · locked

The report is available below and in the public summary.

Review prompt

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

...

for displayed mathematics, with balanced delimiters and braces. Do not leave raw LaTeX commands outside math delimiters. 12. Separate theorem truth from the correctness of the printed proof. A theorem can be Correct in Part 1 because of a fully verified independent proof while its printed proof is adverse in Part 2. Conversely, a defective proof alone never licenses an Incorrect theorem classification. Organize the report under Parts 1–3 and apply Part 4 throughout. Do not add editorial commentary, scores, rankings, recommendations, or a list of suppressed observations.

This panel is visibly marked as owner-only for the concept prototype. Authentication is not enabled locally.

On certain D(9)D(9)D(9) and D(64)D(64)D(64) Diophantine triples

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The claimed classification of all Pellian solutions is false

The D(9)D(9)D(9) Fibonacci-triple classification is not established

The D(64)D(64)D(64) Fibonacci-triple classification is not established

A small-product bound is incorrectly converted into vanishing

One Pell-equation solution class is omitted

The reductions inside the two displayed Pellian classes check out

The factor 1/21/21/2 is missing from the defining unit

Challenge an audit finding

Authenticate with ORCID, then require an exact arXiv match

On certain $D(9)$ and $D(64)$ Diophantine triples

The $D(9)$ Fibonacci-triple classification is not established

The $D(64)$ Fibonacci-triple classification is not established

The factor $1/2$ is missing from the defining unit