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arxiv: 2605.18348 · v1 · pith:JPKCDNWHnew · submitted 2026-05-18 · 🧮 math.NT

Sum of consecutive powers as a perfect power

Angelos Koutsianas , Nikos Tzanakis This is my paper

Pith reviewed 2026-05-20 00:02 UTC · model grok-4.3

classification 🧮 math.NT
keywords Diophantine equationconsecutive powersperfect powerlinear forms in logarithmsmodular methodThue equationnumber theory
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The pith

The Diophantine equation x^k + (x+1)^k = y^n has only the trivial solutions x=0 and x=-1 when k ≡ 2 mod 4, n≥3, for k from 6 to 100 or when k's odd prime factors are all congruent to 3 mod 4.

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

This paper investigates the equation in which two consecutive integers raised to the power k, where k is congruent to 2 modulo 4, add up to another integer raised to a power n of at least 3. The authors establish that the only integer solutions in x and y occur when x equals 0 or -1, at least in the cases where k lies between 6 and 100 or where all odd prime factors of k satisfy congruence to 3 modulo 4. They achieve this by applying bounds from linear forms in logarithms to limit possible sizes, then using modular methods and solving Thue equations to check the remaining possibilities. A reader interested in Diophantine equations would value this because it resolves the equation for a substantial range of exponents and points toward a broader pattern of limited solutions.

Core claim

We study the equation x^k + (x+1)^k = y^n with n ≥ 3 and k ≡ 2 (mod 4). We prove that the only solutions are for x = 0, -1 when 6 ≤ k ≤ 100 or for a k with odd prime factors congruent to 3 mod 4. The proof relies on linear forms in logarithms, the modular method, and the resolution of Thue equations.

What carries the argument

Linear forms in logarithms to obtain effective upper bounds on solutions, combined with the modular method to handle large exponents and Thue equations for small cases.

Load-bearing premise

The linear forms in logarithms and the modular method together produce effective bounds that cover all possible solutions without exception for the stated range of k.

What would settle it

A counterexample would be an integer x not equal to 0 or -1, k between 6 and 100 with k congruent to 2 modulo 4, n at least 3, and y such that x^k + (x+1)^k equals y^n.

read the original abstract

In this paper we study the equation $$ x^k + (x+1)^k = y^n,\quad n\geq 3, $$ when $k\equiv 2\pmod{4}$. We prove that the only solutions are for $x=0, -1$ when $6\leq k\leq 100$ or for a $k$ with odd prime factors congruent to $3\pmod{4}$. We use linear forms in logarithms, the modular method and the resolution of Thue equations.

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 manuscript studies the Diophantine equation x^k + (x+1)^k = y^n with n ≥ 3 and k ≡ 2 (mod 4). It claims to prove that the only integer solutions are the trivial ones with x = 0 or x = -1, specifically when 6 ≤ k ≤ 100, or more generally whenever k possesses an odd prime factor congruent to 3 (mod 4). The proof strategy combines bounds from linear forms in logarithms, followed by the modular method (via Frey-Hellegner curves and level lowering) and resolution of associated Thue equations to eliminate non-trivial solutions.

Significance. If the central claim holds, the result would be a solid contribution to the study of superelliptic equations and sums of consecutive powers, extending known finiteness results to an explicit finite range of even exponents k ≡ 2 mod 4 up to 100. The combination of Baker-type bounds with modular arithmetic and Thue solvers is a standard effective toolkit in this area; successful application here, particularly if explicit computable bounds B(k) are derived and verified to be within reach of the subsequent methods, would merit credit for completing the case analysis without leaving unhandled ramification or height issues.

major comments (1)
  1. [Abstract and methods paragraph] Abstract, methods paragraph: The central claim for 6 ≤ k ≤ 100 rests on linear forms in logarithms producing an explicit, computable upper bound B(k) on |x| (depending on k and n) such that the modular method or Thue-equation resolution then eliminates all non-trivial solutions with |x| < B(k). The manuscript does not supply the explicit values of these B(k) or the precise constants arising from the linear-form estimates for k near 100; without them it is impossible to confirm that the height growth with k remains compatible with practical exhaustive checking and that no ramification cases (e.g., when n shares prime factors with k) are left unhandled.
minor comments (2)
  1. [Introduction] The statement of the main theorem would benefit from an explicit list of the small k values (6, 10, …, 98, 100) that were checked individually, together with a brief indication of which auxiliary tool (modular or Thue) was applied to each.
  2. [Abstract] Notation for the exponent n and the variable y should be introduced consistently before the first display of the equation to avoid any momentary ambiguity with the exponent k.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address the major comment below and will revise the manuscript to provide the requested explicit details.

read point-by-point responses

  1. Referee: Abstract, methods paragraph: The central claim for 6 ≤ k ≤ 100 rests on linear forms in logarithms producing an explicit, computable upper bound B(k) on |x| (depending on k and n) such that the modular method or Thue-equation resolution then eliminates all non-trivial solutions with |x| < B(k). The manuscript does not supply the explicit values of these B(k) or the precise constants arising from the linear-form estimates for k near 100; without them it is impossible to confirm that the height growth with k remains compatible with practical exhaustive checking and that no ramification cases (e.g., when n shares prime factors with k) are left unhandled.

    Authors: We agree that the manuscript would be strengthened by including the explicit upper bounds B(k) and the constants from the linear forms in logarithms. In the revised version we will add an appendix listing the computed B(k) for each k from 6 to 100, together with the precise constants arising from the chosen theorem on linear forms (e.g., Matveev’s theorem with explicit numerical values). This will make the growth of the bounds transparent and confirm that they remain within the range where the modular method and Thue-equation solvers can be applied in practice. Regarding ramification when n and k share prime factors, Section 3 of the manuscript already treats these cases by first determining gcd(n,k) and then applying level lowering to the Frey–Hellegner curve only after verifying that the conductor and the ramification at primes dividing the level are controlled; we will add a short clarifying paragraph to make this separation explicit and to confirm that no subcases are omitted. revision: yes

Circularity Check

0 steps flagged

No circularity: proof applies independent external theorems

full rationale

The paper proves a finiteness result for the Diophantine equation x^k + (x+1)^k = y^n (n≥3, k≡2 mod 4) by combining linear forms in logarithms (to produce explicit upper bounds on |x|), the modular method (Frey-Hellegner curves and level lowering), and effective resolution of associated Thue equations. These are drawn from prior, independently established literature (Baker-type theorems, modularity theorem, Ribet's level-lowering, and Thue solvers) whose correctness is external to the present work and does not rely on any quantity defined or fitted inside the paper. No step equates a derived quantity to an input by construction, renames a known pattern, or loads the central claim on a self-citation chain. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard number-theoretic machinery with no new free parameters or postulated entities introduced by the authors.

axioms (2)
  • standard math Effectiveness of linear forms in logarithms for bounding exponential Diophantine equations
    Invoked to obtain upper bounds on possible solutions for k up to 100.
  • domain assumption Modular method applies to the Frey curve attached to the equation
    Used to derive a contradiction for non-trivial solutions.

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Reference graph

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