arXiv:2606.18500v1

On the Diophantine Inequality $\lvert x^{2} - 2^{a}\cdot 3^{b}\rvert < 3\max\{a,b\}$

Banu İrez Aydın, Herbert Batte, İlker İnam, Florian Luca, Zeynep Demirkol Özkaya

math.NT11D6111J7011J86

Paper metadata

Original submission: June 16, 2026
Claimed MSC: 11D61, 11J70, 11J86
arXiv categories: math.NT
Paper license: CC BY 4.0 ↗
Comments: 17 pages

Abstract

In this paper, we show that there are $57$ nonnegative integer solutions $(a,b,x)$ to the inequality $1\le \lvert x^{2} - 2^{a}\cdot 3^{b}\rvert < 3\max\{a, b\}$ and we list them explicitly. The inequality is converted into a statement about how closely $x/q$ approximates irrational number $\sqrt{d}$ for $d\in\{2,3,6\}$ , where $q$ is an integer which is $3$ -smooth, after which Worley's theorem on rational approximations via continued fractions is applied to parametrise the solutions and a $p$ -adic lower bound for a linear form in logarithms due to Bugeaud and Laurent is applied to find a rather large bound on $\max\{a,b\}$ . We finish with an application of the LLL algorithm to reduce this bound.

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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

01Statements1 reported findingContains unsupported statements

The listed 57 triples are verified throughout the claimed finite search range $\max\{a,b\}\leq2200$ . The assertion that there are no further triples is not able to be verified because the public LLL computation does not cover the parameter range or all coefficient families required by the large-case proof.

Theorem 1.1Not able to verify

The exclusion of all solutions above the search range is not verified

Pages 2 and 14–16 · Theorem 1.1 and Sections 3.3–3.4 · arXiv:2606.18500v1

The finite computation finds exactly the 57 displayed triples when $0\leq a,b\leq2200$ . To conclude that this is the complete global list, the proof requires a uniform LLL lower bound for every admissible coefficient $C$ arising from all six rows of Table 3.1 and from pairs satisfying $rs<2K$ . The cited public notebook instead loops only over $1\leq r,s<400$ and implements one coefficient family for each $d\in\{2,3,6\}$ . Its stored output is not the output of the displayed cell and does not contain the claimed lower bound $2.35\times10^8$ . Thus no counterexample to Theorem 1.1 is established, but the available argument does not exclude solutions with $\max\{a,b\}>2200$ .

02Proofs5 reported findingsContains incorrect or incomplete proofs

The continued-fraction reduction and finite search are supported, but the decisive LLL verification covers only a small subset of the stated cases. The preceding $p$ -adic application also misidentifies an exponent and does not verify the common residue order required by the quoted theorem.

Section 3.1Correct and complete

The finite search verifies the displayed list within $A\leq2200$

Page 7 · Section 3.1 and the cited small-cases notebook · arXiv:2606.18500v1

For fixed $a,b$ , any solution lies among the nearest nonnegative integers to $\sqrt{2^a3^b}$ , with the additional small-value range handled explicitly. Exhausting $0\leq a,b\leq2200$ gives 42 exponent pairs and exactly the 57 triples printed in the table. The candidate selection and the strict inequalities are implemented with exact integer arithmetic.

Application of Theorem 2.3Typo

The chosen base and exponent do not reproduce the linear form

Page 12 · paragraph preceding equation (3.21) · arXiv:2606.18500v1

The paper sets $\eta_1=\gamma^2$ and $b_1=2n$ but then identifies $\eta_1^{b_1}$ with $\gamma^{2n}$ ; the displayed choices actually give $\gamma^{4n}$ . The unique correction is $b_1=n$ . The later estimate using $2n/(\log A+5.5)$ remains a valid upper bound for the corrected $E'$ , so this exponent correction alone does not damage the numerical inequality.

Equation (3.21)Incomplete as written

The common residue order is checked for only one algebraic number

Pages 4 and 12 · Theorem 2.3 and equation (3.21) · arXiv:2606.18500v1

The quoted Bugeaud--Laurent theorem defines $g$ as the smallest positive integer for which both $\eta_1^g-1$ and $\eta_2^g-1$ have positive $p$ -adic valuation. Equation (3.21) computes only the residue order of $\eta_1=\gamma^2$ and never checks the varying number $\eta_2=\varepsilon_1D/C$ . At the primes above $3$ , a unit residue need not have order dividing the value printed in (3.21). Repair classification: Plausible repair only. One may use a uniform multiple of the orders in the finite residue fields, at the cost of enlarging the constants, but the later numerical and LLL bounds would then have to be recomputed.

Section 3.3Incomplete as written

The cited LLL computation does not exhaust the required inputs

Pages 14–15 · equations (3.28) and the paragraph claiming $l(L,y)\geq2.35\times10^8$ · arXiv:2606.18500v1

The proof says the computation scans all admissible coprime $(r,s)$ and every $C$ from the six coefficient families. The cited notebook sets $R_{\max}=400$ , loops only over $r,s<R_{\max}$ , and implements only the first family for each field. The displayed paper permits pairs on a scale up to $rs<2.14A$ with $A$ initially above $7\times10^7$ , and four coefficient families are absent from the notebook. Moreover, the notebook's stored output reports unrelated preliminary bounds rather than its own printed quantities. Repair classification: No repair supplied. A reproducible exhaustive computation or a certified uniform lower-bound argument for all remaining inputs is required; this step is what reduces $A$ from the transcendence bound to $A\leq2200$ .

Sections 3.2 and 3.4Correct and complete

Continued-fraction setup and exceptional cases

Pages 7–16 · Sections 3.2 and 3.4 · arXiv:2606.18500v1

The parity split, normalization by the common $3$ -smooth divisor, Worley parametrization, and Binet formulas lead to the six algebraic coefficient families with the stated elementary size bounds. When the two algebraic numbers are multiplicatively dependent, the lifting-the-exponent argument gives a direct logarithmic bound incompatible with $A>2200$ . These portions do not repair the unverified multiplicatively independent LLL branch.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2606.18500v1
Authors listed: Banu İrez Aydın, Herbert Batte, İlker İnam, Florian Luca, Zeynep Demirkol Özkaya
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

...

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.

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On the Diophantine Inequality ∣x2−2a⋅3b∣<3max⁡{a,b}\lvert x^{2} - 2^{a}\cdot 3^{b}\rvert < 3\max\{a,b\}∣x2−2a⋅3b∣<3max{a,b}

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Abstract

Audit summary

Detailed mathematical audit

The exclusion of all solutions above the search range is not verified

The finite search verifies the displayed list within A≤2200A\leq2200A≤2200

The chosen base and exponent do not reproduce the linear form

The common residue order is checked for only one algebraic number

The cited LLL computation does not exhaust the required inputs

Continued-fraction setup and exceptional cases

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On the Diophantine Inequality $\lvert x^{2} - 2^{a}\cdot 3^{b}\rvert < 3\max\{a,b\}$

The finite search verifies the displayed list within $A\leq2200$