arXiv:2606.28227v1

Perfect powers in sequences of polygonal numbers

Andrzej Dąbrowski, Salah Eddine Rihane, Gökhan Soydan, Paul M. Voutier

math.NT11D4111D5911J8611Y50

Paper metadata

Original submission: June 26, 2026
Claimed MSC: 11D41, 11D59, 11J86, 11Y50
arXiv categories: math.NT
Paper license: CC BY-NC-SA 4.0 ↗
Comments: published version, but comments are still welcome
Journal reference: J. Number Theory v291 (2027)
DOI: 10.1016/j.jnt.2026.04.004 ↗

Abstract

Let $P_s(n)$ denote the $n$ -th $s$ -gonal number. Consider the Diophantine equation $P_{s}(n) = t^{m}$ for integers $n, s, t$ and $m > 2$ . All solutions to this equation are known for $m>2$ and $s\in\{3,5,6,8,10,20\}$ . Here we extend these results to the cases $s = 2k+4$ (where $k = 4,6$ or $5 \leq k \leq 97$ is a prime number) and $s = k+4$ (where $k = 9,15$ or $3 \leq k \leq 97$ is a prime number). The proofs of our results use the modular and hypergeometric methods, linear forms in logarithms and extensive calculations. We were unable to completely solve the above Diophantine equations, but we expect (based on GRH and the weak effective $abc$ conjecture) that there will be no additional solutions beyond those explicitly shown in Theorems~1, 2 and 3.

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 unsupported statements

The explicit fourth- and third-power lists, the exclusions for $p=5,7$ and above the stated logarithmic bounds, and the results explicitly conditioned on GRH or the weak effective $abc$ conjecture are supported in their stated conditional scope. The unconditional uniqueness-and-size assertions in parts (iv) of Theorems 1–3 are not able to be verified from the supplied lemmas.

Theorems 1–3(iv)Not able to verify

The global at-most-one assertion is not established

Pages 2–4 · Theorems 1–3(iv); pages 7–24 · Lemmas 6, 9–10, 13, 17, 19, 23, and case analyses · arXiv:2606.28227v1

The uniqueness inputs, Lemmas 6 and 10, apply only to the generic binomial equations $(k+1)x^p-y^p=k$ and $(k+2)x^p-2y^p=k$ , and only when $p\geq\max\{17,\,2.1\log k+7.5\}.$ Parts (iv) instead begin unconditionally at $p=11$ and concern every solution of the original polygonal-number equation. The other valuation branches lead to different Thue equations. Lemmas 17(c), 19(b), and 23(b) explicitly leave several of those equations unresolved without GRH or the weak effective $abc$ conjecture. No argument shows that a possible solution in an exceptional branch cannot coexist with the one additional solution allowed by Lemma 6 or 10, and no $10^{100000p}$ size estimate is proved for all those branches. Thus neither the global at-most-one conclusion nor its universal size bound follows as stated.

Full paper, version 1 ↗

Theorems 1–3(vi)Minor formal correction

The trivial zero solution must remain in the conditional conclusion

Pages 2–4 · Theorems 1–3(vi) · arXiv:2606.28227v1

For every displayed polygonal equation, $(n,t)=(0,0)$ remains a solution for every prime exponent. Parts (vi) say that $(1,1)$ is the only solution under the weak effective $abc$ conjecture. The intended and uniquely determined statement is that $(1,1)$ is the only nonzero solution, or equivalently that the only solutions are $(0,0)$ and $(1,1)$ . This endpoint correction does not affect any nontrivial Diophantine claim or the overall status beyond the separate unsupported part (iv).

Full paper, version 1 ↗

Theorems 1–3(iv)Minor formal correction

The lower bound should be on $|n|$

Pages 2–4 · Theorems 1–3(iv); pages 10–13 · continued-fraction lemmas · arXiv:2606.28227v1

The continued-fraction calculation gives a lower bound for $|x|$ , and the paper expressly allows negative polygonal indices. In the generic factorization, $n=x^p$ , so the immediate consequence is $|n|\geq10^{100000p}$ , not $n\geq10^{100000p}$ . Replacing $n$ by $|n|$ is a local range correction. It does not repair the independent coverage gaps in part (iv), but it is the correct form of the bound wherever the continued-fraction lemma applies.

Full paper, version 1 ↗

02Proofs3 reported findingsContains incorrect or incomplete proofs

The proof assembly for parts (iv) extends generic uniqueness and continued-fraction lemmas beyond their stated exponent ranges and beyond the factorization branches they treat. Two weak-effective- $abc$ applications are written with a negative summand, although those local steps admit a direct positive-triple repair.

Proof of Theorems 1–3(iv)Incomplete as written

The cited lemmas do not cover the full exponent range or all valuation branches

Pages 4, 7–13, and 15–24 · proof roadmap, Lemmas 6, 9–10, 13, 17, 19, 23 · arXiv:2606.28227v1

The introduction says parts (iv) follow immediately from Sections 4 and 5. Those sections analyze only the generic equations and their uniqueness lemma starts at $\max\{17,2.1\log k+7.5\}$ , not at $11$ . The remaining sections produce separate equations for $k\mid n$ and $k^2\mid n$ and leave named exceptional pairs unresolved unconditionally. Those branches are not brought under the uniqueness theorem or the continued-fraction size bound. Repair classification: No repair supplied for parts (iv) as stated. A verified scope repair would restrict the assertion to solutions arising from the generic branch and to exponents satisfying the hypotheses of Lemmas 6 or 10 and the documented continued-fraction calculation.

Full paper, version 1 ↗

Proof of Lemma 9Incomplete as written

The documented computation starts at $p=17$ while the lemma starts at $p=7$

Pages 11–12 · Lemma 9 and proof · arXiv:2606.28227v1

Lemma 9 is stated for every prime $7\leq p\leq P_{k,1001}$ , but its proof says that the continued fractions were calculated for $17\leq p\leq P_{k,1001}$ . The earlier brute-force search only covers $|y|\leq1000$ and therefore does not fill the interval up to $10^{100000}$ . In particular, the proof as printed does not establish the large-denominator exclusion for $p=11$ or $13$ , which are within the range used by part (iv). Repair classification: No missing computation or separate argument is supplied in the paper.

Full paper, version 1 ↗

Proofs of Lemmas 17(d) and 23(c)Incorrect as written

The weak effective $abc$ conjecture is applied with a negative summand

Pages 17 and 20–21 · proofs of Lemmas 17(d) and 23(c) · arXiv:2606.28227v1

The paper defines $abc(r)$ for relatively prime positive integers $a+b=c$ , but these proofs take $b=-1$ . That substitution is outside the stated conjecture. Repair classification: Verified repair. Nonzero solutions have the two variables of the same sign. For positive variables, rewrite $A x^p-y^p=1$ as $y^p+1=A x^p$ ; for negative variables, write $A X^p+1=Y^p$ . The radical is still bounded by the same product of the fixed coefficients and the two variables, and the displayed exponent ranges give the required lower bound for $L(a,b,c)$ . Thus the conditional exclusions can be recovered, but not by the signed triple printed in the proof.

Full paper, version 1 ↗

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2606.28227v1
Authors listed: Andrzej Dąbrowski, Salah Eddine Rihane, Gökhan Soydan, Paul M. Voutier
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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Perfect powers in sequences of polygonal numbers

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The global at-most-one assertion is not established

The trivial zero solution must remain in the conditional conclusion

The lower bound should be on ∣n∣|n|∣n∣

The cited lemmas do not cover the full exponent range or all valuation branches

The documented computation starts at p=17p=17p=17 while the lemma starts at p=7p=7p=7

The weak effective abcabcabc conjecture is applied with a negative summand

Challenge an audit finding

Authenticate with ORCID, then require an exact arXiv match

The lower bound should be on $|n|$

The documented computation starts at $p=17$ while the lemma starts at $p=7$

The weak effective $abc$ conjecture is applied with a negative summand