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arxiv: 2606.28227 · v1 · pith:HT5RWKI4new · submitted 2026-06-26 · 🧮 math.NT

Perfect powers in sequences of polygonal numbers

Andrzej D\k{a}browski , Salah Eddine Rihane , G\"okhan Soydan , Paul M. Voutier This is my paper

Pith reviewed 2026-06-29 02:20 UTC · model grok-4.3

classification 🧮 math.NT
keywords polygonal numbersperfect powersDiophantine equationsmodular methodhypergeometric identitieslinear forms in logarithmss-gonal numbersperfect power equation
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The pith

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.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

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.

Core claim

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.

What carries the argument

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.

If this is right

  • 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.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • 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.

Load-bearing premise

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.

What would settle it

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.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

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.

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 (1)
  1. [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.
minor comments (2)
  1. [Abstract] 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.
  2. [Abstract and §1] 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.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment of significance, and recommendation of minor revision. We address the single major comment below.

read point-by-point responses

  1. Referee: [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.

    Authors: 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: yes

Circularity Check

0 steps flagged

No significant circularity; external methods and explicit conditional statement

full rationale

The paper applies the modular method, hypergeometric identities, linear forms in logarithms, and computations to locate solutions to P_s(n) = t^m. Theorems 1-3 present the explicit solutions found. The abstract directly states that the equations could not be completely solved and that completeness is expected only conditionally on GRH and the weak effective abc conjecture. These are independent external conjectures, not internal reductions. No self-definitional steps, fitted inputs renamed as predictions, load-bearing self-citations, or ansatz smuggling appear in the derivation chain. The work is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard number-theoretic machinery without introducing new free parameters or postulated entities.

axioms (2)
  • standard math Standard bounds and effectiveness results from the theory of linear forms in logarithms
    Invoked to obtain upper bounds on possible solutions.
  • domain assumption Applicability of the modular method to the polygonal-power equation for the given s
    Assumed to produce useful congruences that restrict solutions.

pith-pipeline@v0.9.1-grok · 5750 in / 1303 out tokens · 66803 ms · 2026-06-29T02:20:46.699895+00:00 · methodology

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Reference graph

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