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arxiv: 2604.25017 · v1 · submitted 2026-04-27 · 🧮 math.NT

Ramanujan, the taxicab problem for polynomials, and the abc-conjecture

Katalin Gyarmati This is my paper

Pith reviewed 2026-05-08 01:19 UTC · model grok-4.3

classification 🧮 math.NT
keywords Ramanujantaxicab problempolynomialsabc-conjecturepower sumsDiophantine equationssolvability
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The pith

The solvability of power-sum equations like p^n + q^n = r^n + s^n for polynomials connects to the abc-conjecture.

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

Ramanujan's taxicab problem, finding numbers equal to two different sums of two cubes, is generalized here to polynomials. The paper investigates for which n and polynomials p, q, r, s the equation holds, and extends to the more general case of several powered polynomials summing to zero. It proposes using the abc-conjecture in its polynomial form to determine when non-trivial solutions exist or do not. This provides a bridge between classical Diophantine problems and their analogs in algebra.

Core claim

Starting with Ramanujan's famous taxicab problem, the solvability of the equations p^n + q^n = r^n + s^n and, more generally, p_1^{k_1} + … + p_m^{k_m} = 0 among polynomials can be studied by relating them to the polynomial analog of the abc-conjecture.

What carries the argument

The generalization of the taxicab equation to polynomials and its connection to the abc-conjecture for bounding or proving existence of solutions.

Load-bearing premise

That the abc-conjecture provides useful information about the solvability of these specific polynomial power sum equations.

What would settle it

Discovery of a family of non-constant polynomials satisfying p^n + q^n = r^n + s^n for arbitrarily large n, which would contradict predictions from the abc-conjecture if it implies no such solutions exist.

read the original abstract

Starting with Ramanujan's famous taxicab problem, we can study the solvability of the equations $p^n+q^n=r^n+s^n$ and, more generally, $p_1^{k_1}+\dots+p_m^{k_m}=0$ among polynomials.

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 / 1 minor

Summary. The manuscript starts from Ramanujan's taxicab number 1729 and proposes to study the solvability of equations of the form p^n + q^n = r^n + s^n together with the more general power-sum equation p_1^{k_1} + ⋯ + p_m^{k_m} = 0 when the variables are polynomials rather than integers.

Significance. If concrete non-solvability results or explicit polynomial solutions were derived and shown to follow from the polynomial abc-conjecture (which is known to hold in several settings), the work could supply function-field analogues of classical Diophantine statements. The present text, however, contains no theorems, examples, or derivations, so any potential significance remains prospective.

major comments (1)
  1. Abstract: the central claim that the solvability question 'can be studied' among polynomials is not supported by any theorem, example, or derivation; the manuscript therefore provides no load-bearing mathematical content against which the suggested link to the abc-conjecture can be evaluated.
minor comments (1)
  1. The abstract is the only text supplied; a full manuscript should include at least one explicit polynomial identity or non-identity together with a statement of the precise ring and degree constraints under consideration.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. We address the major comment below.

read point-by-point responses

  1. Referee: Abstract: the central claim that the solvability question 'can be studied' among polynomials is not supported by any theorem, example, or derivation; the manuscript therefore provides no load-bearing mathematical content against which the suggested link to the abc-conjecture can be evaluated.

    Authors: We agree that the manuscript, as written, contains no theorems, examples, or derivations. It is a short announcement proposing that the solvability of equations such as p^n + q^n = r^n + s^n and more general power-sum equations can be studied when the variables are polynomials, and suggesting a possible connection to the abc-conjecture in the function-field setting. The referee correctly notes that this leaves any significance prospective. We will revise the manuscript to include at least one explicit polynomial example (or a non-solvability result drawn from a known case of the polynomial abc-conjecture) so that the suggested link can be evaluated against concrete content. revision: yes

Circularity Check

0 steps flagged

No significant circularity; exploratory discussion only

full rationale

The manuscript poses solvability questions for polynomial power-sum equations (e.g., p^n + q^n = r^n + s^n) inspired by Ramanujan's taxicab problem and notes a possible connection to the polynomial abc-conjecture. No derivation, theorem, or quantitative prediction is advanced whose validity reduces to a fitted parameter, self-definition, or load-bearing self-citation. The text contains no equations that equate a claimed result to its own inputs by construction, no ansatz smuggled via prior work, and no uniqueness theorem invoked from the author's own earlier papers. The activity is therefore self-contained as an open-ended exploration rather than a closed deductive chain.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Insufficient information; only the abstract is provided, so no free parameters, axioms, or invented entities can be identified from the text.

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Reference graph

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