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arxiv: 2606.07735 · v1 · pith:UJTBXGQVnew · submitted 2026-06-05 · 🧮 math.NT · hep-ph· hep-th· math-ph· math.MP

Arithmetic Symmetry in Ideal Prouhet-Tarry-Escott Solutions

Yu-Dai Tsai , Junseok Lee , Fuminobu Takahashi This is my paper

Pith reviewed 2026-06-27 20:42 UTC · model grok-4.3

classification 🧮 math.NT hep-phhep-thmath-phmath.MP
keywords Prouhet-Tarry-Escott problemsum of two squaressymmetric solutionsasymptotic countingDiophantine equationsinteger representations
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The pith

The symmetric locus in the ideal degree-three Prouhet-Tarry-Escott problem reduces exactly to the sum-of-two-squares equation and is counted by (4 log 2 / 3 π²) H³ log H + O(H³).

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

The paper studies symmetric integer solutions to the ideal degree-three Prouhet-Tarry-Escott problem, in which the entries are paired symmetrically about a common center. This pairing reduces the defining equations to the single Diophantine condition x² + y² = u² + v² together with parity constraints on the variables. The resulting count N_sym(H) of solutions of bounded height therefore inherits the second-moment asymptotics of the sum-of-two-squares representation function, producing a logarithmic factor beyond the naive cubic growth. The existence of this large arithmetically structured subfamily shows that the full solution space contains a substantial component governed by classical arithmetic rather than by generic volume estimates alone.

Core claim

For the full symmetric locus, N_sym(H) equals (4 log 2 / 3 π²) H³ log H + O(H³); the logarithmic enhancement arises directly from the second moment of the sum-of-two-squares representation function, and the symmetric solutions therefore form a large arithmetically structured subfamily of the ideal degree-three solution space.

What carries the argument

The reduction of symmetric ideal degree-three PTE equations to the sum-of-two-squares equation x² + y² = u² + v² subject to the stated parity conditions.

If this is right

  • The number of symmetric solutions exceeds the naive H³ box-counting scale by a logarithmic factor.
  • The entire symmetric locus is controlled by the arithmetic of representations as sums of two squares.
  • This subfamily supplies an explicit, infinite collection of ideal degree-three solutions whose size is governed by classical number theory.
  • Paired anomaly-free integral charge spectra are realized inside a number-theoretically natural subset of the solution space.

Where Pith is reading between the lines

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

  • The same symmetry reduction may produce analogous logarithmic enhancements in higher-degree PTE problems.
  • Other structured subfamilies of the full PTE solution space could be located by imposing additional linear or modular conditions.
  • The asymptotic supplies a concrete benchmark against which exhaustive computer searches for small-height solutions can be validated.

Load-bearing premise

The symmetry condition reduces the ideal degree-three PTE equations exactly to the sum-of-two-squares equation subject only to the stated parity conditions, with no further obstructions that would alter the leading asymptotic.

What would settle it

An exact enumeration of all symmetric solutions with height up to several thousand that deviates from the predicted leading coefficient times H³ log H by more than the O(H³) term.

Figures

Figures reproduced from arXiv: 2606.07735 by Fuminobu Takahashi, Junseok Lee, Yu-Dai Tsai.

Figure 1
Figure 1. Figure 1: Finite-height enumeration of nontrivial ideal degree-three Prouhet– Tarry–Escott solutions of size four. The blue markers give the full symmetric count Nsym(H), including both center classes c ∈ Z and c ∈ Z + 1 2 , while the orange markers give the non-symmetric count Nnon-sym(H). All counts use the multiset, interchange, and admissible-center conventions described in the text. The gray dashed curve is inc… view at source ↗
Figure 2
Figure 2. Figure 2: Observed symmetric proportion R(H) among all nontrivial ideal degree-three Prouhet–Tarry–Escott solutions of size four in the finite enumera￾tion, counted with the multiset, interchange, and admissible-center conventions of the text. The proportion decreases over the displayed range but remains substantial there. Appendix C. Alternative Counting Geometry: Finite-Squared-Sum Enumeration In the main body of … view at source ↗
Figure 3
Figure 3. Figure 3: Centered nontrivial symmetric solutions ordered by the one-side centered squared-sum cutoff U. The dashed curve corresponds to (C.38). This should be compared with the height-box asymptotic Csym(H) = 2 log 2 π 2 H2 log H + O(H2 ). (C.39) Both counts are governed by the same arithmetic second moment of r2(n), but the geometric averaging is different. The radial cutoff orders representations directly by the … view at source ↗
read the original abstract

Motivated in part by anomaly cancellation for integral charge spectra in chiral gauge theory, we study the symmetric locus in the ideal degree-three Prouhet-Tarry-Escott problem. A symmetric integer solution is one whose entries are paired about a common center $c\in \frac12\mathbb Z$. This symmetry reduces the problem to a sum-of-two-squares equation, $x^2+y^2=u^2+v^2$, in integer variables, subject to the appropriate parity conditions. Thus the problem is governed by representations as sums of two squares. For the full symmetric locus, let $N_{\mathrm{sym}}(H)$ denote the number of nontrivial symmetric integer solutions of height at most $H$, counted with unordered multiset conventions and summed over the admissible centers. Then \begin{align*} N_{\mathrm{sym}}(H) = \frac{4\log 2}{3\pi^2}H^3\log H+O(H^3). \end{align*} The logarithmic enhancement comes from the second moment of the sum-of-two-squares representation function. In particular, the symmetric locus is larger than one would expect from the naive $H^3$ degree-weighted box-counting scale alone. This asymptotic identifies a large arithmetically structured subfamily of the ideal degree-three solution space, and suggests that paired anomaly-free integral charge spectra reflect a fundamental number-theoretic structure.

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

0 major / 2 minor

Summary. The manuscript studies the symmetric locus of ideal degree-three Prouhet-Tarry-Escott solutions. It shows that pairing entries about a common center reduces the ideal PTE conditions exactly to the sum-of-two-squares equation x² + y² = u² + v² subject to parity constraints. It then counts the nontrivial symmetric solutions of height at most H (with unordered multiset conventions, summed over admissible centers) and derives the asymptotic N_sym(H) = (4 log 2 / 3 π²) H³ log H + O(H³), with the logarithmic factor arising from the second moment of the sum-of-two-squares representation function r_2.

Significance. If the reduction and counting argument hold, the result identifies a large arithmetically structured subfamily of ideal degree-three solutions whose size exceeds the naive H³ scale by a log H factor. The link to anomaly cancellation in chiral gauge theory is noted but not developed. The derivation rests on standard analytic number theory (second-moment asymptotics for r_2) rather than ad-hoc fitting, which is a strength.

minor comments (2)
  1. [§2] The abstract and introduction state the reduction to x² + y² = u² + v² but do not display the explicit parity conditions or the cancellation of odd-powered sums; a short displayed derivation in §2 would improve readability.
  2. [§4] The error term O(H³) absorbs diagonal, boundary, and distinctness contributions; a one-sentence remark on why these are indeed O(H³) rather than larger would clarify the argument without lengthening the paper.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the recognition of the reduction to the sum-of-two-squares equation and the derivation of the asymptotic via the second moment of r_2. We appreciate the recommendation to accept.

Circularity Check

0 steps flagged

No circularity; derivation applies known external asymptotic to a self-contained reduction

full rationale

The paper reduces the symmetric PTE locus to the equation x² + y² = u² + v² under explicit parity conditions on the variables, then invokes the classical second-moment asymptotic ∑_{m≤X} r₂(m)² ∼ c X log X (with c = 4 log 2 / π² or an equivalent normalization) to obtain the stated count N_sym(H). This second-moment result is a standard theorem in analytic number theory, independent of the present work and externally verifiable. No parameter is fitted to the target count, no self-citation supplies the leading term, and the reduction itself is derived directly from the symmetry assumption without presupposing the final asymptotic. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The sole structural assumption is the exact reduction of the symmetric PTE equations to the sum-of-two-squares problem; no free parameters, new entities, or additional axioms are introduced in the abstract.

axioms (1)
  • domain assumption Symmetric solutions reduce exactly to the equation x² + y² = u² + v² subject to the appropriate parity conditions.
    This reduction is asserted in the abstract as the step that makes the problem amenable to sums-of-two-squares counting.

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discussion (0)

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Reference graph

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