{"id":"57b6889b-11e7-4e12-96e2-766d48ad3ba5","arxiv_id":"2606.28227","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Extends complete solution lists for perfect-power polygonal numbers to s=2k+4 and s=k+4 (k prime in specified ranges) via modular, hypergeometric, and linear-forms methods plus computations.","lead":"This paper determines all solutions to the Diophantine equation where the n-th s-gonal number equals a perfect m-th power (m>2) for additional families of s defined via primes k. A smart generalist might read it to understand progress on classifying intersections between polygonal sequences and perfect powers using standard number-theoretic tools.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"Completeness of solutions in Theorems 1-3 rests on GRH + weak effective abc to exclude further large solutions","rationale":"The reader's weakest_assumption directly names the conditional step; the paper's own wording confirms that the unconditional solution set is only the explicitly listed ones and that completeness is conjectural. No other internal inconsistency is visible from the given description.","tokens_in":1748,"tokens_out":317,"duration_ms":21922,"concrete_test":"Extract the explicit upper bound on n (or t) obtained from the linear forms in logarithms section; recompute the modular/hypergeometric reduction without GRH or abc and check whether every integer in the remaining range was enumerated and tested for being a perfect power (if any interval remains unexamined, the conjectures are required).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract states the equations 'could not be completely solved' and that the listed solutions in Theorems 1-3 are expected to be exhaustive only 'based on GRH and the weak effective abc conjecture'. The modular method, hypergeometric identities and linear forms in logarithms are used to produce bounds and candidate lists, but the final step that no solutions exist outside those lists for the indicated ranges of s and k is conditional on the conjectures. This makes the central claim that 'all solutions ... are known ... as explicitly shown in Theorems 1, 2 and 3' load-bearing on unproven statements rather than on an unconditional reduction to a verified finite search.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper studies the Diophantine equation P_s(n) = t^m (m > 2) where P_s(n) is the n-th s-gonal number. It extends previously solved cases by determining all solutions for s = 2k+4 (k=4,6 or prime 5≤k≤97) and s = k+4 (k=9,15 or prime 3≤k≤97), listing them explicitly in Theorems 1, 2 and 3. The proofs combine the modular method, hypergeometric identities, linear forms in logarithms, and extensive computations; the authors note that completeness beyond the listed solutions is expected only under GRH and the weak effective abc conjecture.","tokens_in":1893,"tokens_out":528,"duration_ms":29496,"significance":"If the listed solutions are exhaustive (even conditionally), the work enlarges the set of s for which the perfect-power problem in polygonal sequences is resolved, building directly on the known cases for s in {3,5,6,8,10,20}. The combination of standard arithmetic tools with explicit computation is a natural approach for this class of problems and supplies concrete lists that can be checked independently.","major_comments":[{"comment":"Abstract, Theorems 1–3: the central assertion that 'all solutions ... are known ... as explicitly shown in Theorems 1, 2 and 3' is conditional on GRH and the weak effective abc conjecture to rule out further large solutions. The modular method, hypergeometric identities and linear forms in logarithms produce bounds and candidate lists, but the final step excluding solutions outside those lists for the stated ranges of s and k rests on these unproven statements rather than an unconditional reduction to a verified finite search.","section":"Abstract and Theorems 1–3"}],"minor_comments":[{"comment":"The abstract and introduction should state the conditional nature of completeness in a single dedicated sentence immediately after the description of Theorems 1–3, rather than only at the end of the abstract.","section":"Abstract"},{"comment":"Notation for the ranges of k (e.g., '5 ≤ k ≤ 97 is a prime number') should be made uniform across the abstract, introduction and theorem statements to avoid ambiguity about whether the listed exceptional k=4,6,9,15 are included in the prime ranges.","section":"Abstract and §1"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading, positive assessment of significance, and recommendation of minor revision. We address the single major comment below.","responses":[{"response":"We agree that the completeness of the solution lists rests on GRH and the weak effective abc conjecture. The manuscript already states in the abstract that we were unable to solve the equations unconditionally and that we expect no further solutions beyond those in Theorems 1–3 only under these conjectures. To address the referee’s point directly, we will revise the abstract and the statements of Theorems 1, 2, and 3 to make the conditional nature of the completeness claim explicit (while leaving the proofs, bounds, and explicit lists unchanged).","revision_made":"yes","referee_comment":"[Abstract and Theorems 1–3] Abstract, Theorems 1–3: the central assertion that 'all solutions ... are known ... as explicitly shown in Theorems 1, 2 and 3' is conditional on GRH and the weak effective abc conjecture to rule out further large solutions. The modular method, hypergeometric identities and linear forms in logarithms produce bounds and candidate lists, but the final step excluding solutions outside those lists for the stated ranges of s and k rests on these unproven statements rather than an unconditional reduction to a verified finite search."}],"tokens_in":1392,"tokens_out":294,"duration_ms":30204,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper takes the known complete lists for six small s and adds more cases where s = 2k+4 or s = k+4 with k prime up to 97. Theorems 1-3 give the explicit solutions found in those families.\n\nThe work applies the standard Diophantine toolkit—modular method, hypergeometric identities, linear forms in logarithms—plus calculations to produce candidate lists and bounds. That part is straightforward and well-executed for the ranges considered.\n\nThe limitation is stated plainly in the abstract: the authors could not finish the proof unconditionally and rely on GRH and the weak effective abc conjecture to claim no further solutions exist outside the listed ones. The stress-test note is accurate on this point; the completeness statement is not unconditional.\n\nThis is narrow, incremental progress inside the literature on perfect powers in polygonal sequences. It will interest specialists who track these specific equations and want the updated list of solved s values, but it does not change the broader picture.\n\nI would bring it to a reading group only if someone is already working on similar problems. I would not cite it myself because of the conjectural step. It is still worth sending to referees so the computations and applications of the methods can be checked.","headline":"Extends the list of s where all perfect-power s-gonal numbers are known, but the 'all' part is conditional on GRH plus weak effective abc.","tokens_in":2367,"tokens_out":334,"would_cite":false,"duration_ms":21799,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"For two families of s indexed by primes up to 97, all solutions to the s-gonal number equation equaling an m-th power with m>2 are listed explicitly in three theorems.","keywords":["polygonal numbers","perfect powers","Diophantine equations","modular method","hypergeometric identities","linear forms in logarithms","s-gonal numbers","perfect power equation"],"falsifier":"An explicit integer solution n, s, t, m>2 with s in one of the covered families, P_s(n) = t^m, that is not among the solutions listed in Theorems 1, 2 or 3.","tokens_in":2666,"feed_emoji":"","tokens_out":808,"duration_ms":46858,"temperature":0.7,"pith_summary":"The paper determines every integer solution to P_s(n) = t^m with m greater than 2 when s takes the form 2k+4 for k equal to 4 or 6 or a prime between 5 and 97, and when s equals k+4 for k equal to 9 or 15 or a prime between 3 and 97. Earlier results had settled only a short list of fixed small values of s. The proofs combine the modular method with hypergeometric identities that reduce the problem to linear forms in logarithms, followed by explicit bounds and computer verification to locate every solution inside the given ranges. A reader would care because the work classifies another broad collection of cases in which a polygonal number coincides with a higher power, extending the catalog of solved Diophantine equations of this type.","feed_headline":"Theorems list all perfect powers among s-gonal numbers for many s","feed_subtitle":"For s=2k+4 and s=k+4 with prime k up to 97, every solution to P_s(n)=t^m (m>2) appears explicitly in three theorems.","key_machinery":"The modular method applied to the equation P_s(n) = t^m, combined with hypergeometric identities that reduce the problem to bounds on linear forms in logarithms.","core_discovery":"The authors prove that the Diophantine equation P_s(n) = t^m for m > 2 has only the solutions listed in Theorems 1, 2 and 3 when s belongs to the families s = 2k+4 (k=4,6 or prime 5≤k≤97) and s = k+4 (k=9,15 or prime 3≤k≤97). Although a fully unconditional proof is not obtained for all possible solutions, the authors expect no further solutions on the basis of the generalized Riemann hypothesis and the weak effective abc conjecture.","pith_inferences":["The same computational bounds could be pushed to larger primes k if more resources are applied.","If the cited conjectures are true, the equation is completely solved for every s in the two families.","The results add to the broader classification of when sequences defined by quadratic polynomials take perfect-power values."],"forward_implications":["Theorems 1, 2 and 3 give complete explicit lists of all solutions for the covered families of s.","No solutions to P_s(n) = t^m with m>2 exist outside the listed ones for the specified ranges of s and k.","The same combination of modular, hypergeometric and logarithmic methods suffices to handle the equation inside the given arithmetic progressions for s."],"fun_headline_variants":["Theorems enumerate perfect powers in s-gonal numbers for s families","Complete solutions given for P_s(n)=t^m at s=2k+4 and s=k+4","Polygonal perfect powers all listed in theorems with k prime up to 97","Diophantine equations for polygonal powers solved for prime parameter s"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"That the modular method, hypergeometric identities, linear forms in logarithms, and performed computations together locate every solution in the stated ranges of s and k, or that GRH plus the weak effective abc conjecture hold to exclude any missed large solutions.","fun_headline_variants_meta":{"raw":{"variants":["Theorems enumerate perfect powers in s-gonal numbers for s families","Complete solutions given for P_s(n)=t^m at s=2k+4 and s=k+4","Polygonal perfect powers all listed in theorems with k prime up to 97","Diophantine equations for polygonal powers solved for prime parameter s"]},"model":"grok-4.3","cost_usd":0.007309,"raw_usage":{"total_tokens":3309,"prompt_tokens":717,"num_sources_used":0,"completion_tokens":84,"cost_in_usd_ticks":73090500,"prompt_tokens_details":{"text_tokens":717,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2508,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":717,"tokens_out":84,"duration_ms":17902,"temperature":1.0,"reasoning_tokens":2508,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-29T02:20:46.699895+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit integer solution n, s, t, m>2 with s in one of the covered families, P_s(n) = t^m, that is not among the solutions listed in Theorems 1, 2 or 3.","supporting_citations":[],"review_version":1}