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On zero-sum problems of new types

T0 review · 1 major / 0 minor · reviewed 2026-07-03 · grok-4.3

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Pith's one-line read Given 2n-1 integers not divisible by n>1, some nonempty subset I with |I|≤n has sum divisible by n but not by n².

desk verdict Sun's paper states a clean but modest refinement of zero-sum theorems, adding the condition that the subset sum is divisible by n but not n². read the letter →

arxiv 2606.18234 v5 pith:JK5SHLR2 submitted 2026-06-16 math.NT math.CO

Zhi-Wei Sun This is my paper · ORCID
classification math.NTmath.CO
keywords zero-sumproblemssubsetsumsdivisibilitymodulararithmeticnumbertheoryErdős–Ginzburg–Zivtheoremopenconjectures
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper studies zero-sum problems that impose an extra condition on the exact power of n dividing the subset sum. It establishes that from any 2n-1 integers each not divisible by a fixed n>1, one can always find a nonempty subset of size at most n whose sum meets the exact-divisibility requirement. A reader would care because the result refines the classical zero-sum theorems by adding control over divisibility by n versus n². The work presents this as one concrete instance of new-type zero-sum questions and lists several open conjectures for further cases.

What carries the argument

The existence of a bounded-size subset whose sum is 0 mod n but nonzero mod n², under the hypothesis that no single term is 0 mod n.

What would settle it

A collection of 2n-1 integers, none divisible by some fixed n>1, in which every nonempty subset of size at most n has a sum that is either nonzero mod n or zero mod n².

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Extended reading notes

Core claim

Given 2n-1 integers a1,…,a2n-1 not divisible by an integer n>1, there exists a nonempty I⊆{1,…,2n-1} with |I|≤n such that ∑i∈I ai is divisible by n but not divisible by n². This statement is proved as an example of zero-sum problems of new types, and the paper poses several additional conjectures on related questions.

Load-bearing premise

Each of the 2n-1 integers is not divisible by n.

Editorial extensions

If this is right

  • The result holds for every integer n greater than 1.
  • The subset I is guaranteed to be nonempty and of cardinality at most n.
  • The sum condition distinguishes exact first-power divisibility by n from higher powers.
  • The same style of refined zero-sum statement can be posed for other choices of subset-size bounds or modulus powers.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The new exact-divisibility condition might be combined with the classical Erdős–Ginzburg–Ziv theorem to produce simultaneous statements about sums mod n and mod n².
  • One could check computationally whether the size bound |I|≤n remains valid when the hypothesis is relaxed to allow some ai divisible by n.
  • The conjectures listed in the paper may connect this divisibility refinement to problems about multiple subset sums or different rings.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 0 minor

Summary. The manuscript investigates zero-sum problems of new types in combinatorial number theory. It claims to prove that for any integer n>1 and any 2n-1 integers a1,...,a2n-1 each not divisible by n, there exists a nonempty subset I of {1,...,2n-1} with |I|≤n such that the sum over I is divisible by n but not by n². The paper also poses several conjectures for further research.

Significance. If the claimed theorem holds, it refines the classical Erdős–Ginzburg–Ziv theorem by adding a condition that the subsum has exact n-adic valuation 1 (under the hypothesis that no individual ai is divisible by n) and relaxes the subset size from exactly n to at most n. This could contribute to the study of zero-sums with controlled divisibility properties. The posed conjectures may open avenues for further work in the area.

major comments (1)
  1. [Abstract / main theorem statement] The manuscript asserts the existence of a proof for the main theorem stated in the abstract but provides no derivation, argument, or detailed reasoning anywhere in the text. Without the argument it is impossible to verify correctness, check for gaps, or assess whether the bound |I|≤n is tight or whether the non-divisibility by n² follows directly from the hypothesis.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report and recommendation for major revision. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract / main theorem statement] The manuscript asserts the existence of a proof for the main theorem stated in the abstract but provides no derivation, argument, or detailed reasoning anywhere in the text. Without the argument it is impossible to verify correctness, check for gaps, or assess whether the bound |I|≤n is tight or whether the non-divisibility by n² follows directly from the hypothesis.

    Authors: We agree with the referee that the submitted manuscript states the main theorem but omits the detailed proof, argument, and any discussion of the bound or the n² condition. This was an error in manuscript preparation. In the revised version we will include a complete proof of the stated result together with verification that |I|≤n is sharp and that the sum is not divisible by n² under the given hypotheses. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; direct existence proof under explicit hypothesis

full rationale

The paper states an existence theorem whose hypothesis (each a_i not divisible by n) is explicitly required for the conclusion (a subset sum divisible by n but not n² with |I|≤n) and is not derived from the result itself. No equations, fitted parameters, self-citations, or ansatzes appear in the abstract or described claim; the result is presented as a combinatorial proof rather than a reduction to prior fitted data or self-referential definitions. The derivation chain is therefore self-contained against external benchmarks and does not reduce any prediction to its inputs by construction.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The result rests on standard divisibility and subset-sum counting arguments in elementary number theory; no free parameters, invented entities, or nonstandard axioms are visible from the abstract.

assumptions (1)
  • standard math Basic arithmetic properties of integers and modular arithmetic (divisibility, residues modulo n and n²)
    Invoked implicitly to define the conditions a_i ≢ 0 mod n and sum ≡ 0 mod n but sum ≢ 0 mod n².

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Cite this review

Pith. "Pith review of On zero-sum problems of new types." pith.science (2026). https://pith.science/paper/JK5SHLR2

@misc{pith2026260618234,
  author       = {Pith},
  title        = {Pith review of: On zero-sum problems of new types},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JK5SHLR2}},
  note         = {Machine review of arXiv:2606.18234}
}
abstract

In this paper, we investigate zero-sum problems of new types. For example, given $2n-1$ integers $a_1,\ldots,a_{2n-1}$ not divisible by an integer $n>1$, we prove that for some nonempty $I\subseteq\{1,\ldots,2n-1\}$ with $|I|\leqslant n$, the sum $\sum_{i\in I}a_i$ is divisible by $n$ but not divisible by $n^2$. We also pose several conjectures for further research.

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Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

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    Reiher,On Kemnitz’ conjecture concerning lattice-points in the plane, Ramanujan J.13(2007), 333–337

    C. Reiher,On Kemnitz’ conjecture concerning lattice-points in the plane, Ramanujan J.13(2007), 333–337. ON ZERO-SUM PROBLEMS OF NEW TYPES 19 School of Mathematics, Nanjing University, Nanjing 210093, People’s Re- public of China Email address:zwsun@nju.edu.cn

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