Proofs and disproofs of some conjectures on zero-sum problems of new types
Abstract
For integers r≥1 and n≥2, let sr(n) be the least N such that every sequence of N vectors in Zr, each nonzero modulo n, contains n terms whose sum is zero modulo n but nonzero modulo n2. Let wr(n) be the corresponding least N when any nonempty subsequence is allowed. These constants were introduced as zero-sum problems of new types by Z.-W. Sun. We prove sr(n)≥(2r+1−2)(n−1)+1 by constructing an explicit sequence from a lattice polytope, \emph{disproving} his conjecture s2(n)=4n+1 for n≥4. We obtain wr(2)=2r+1 by classifying all sequences avoiding such nonempty subsequences modulo $4$. We also \emph{prove} Sun's conjecture w2(n)=4n−3 for n=2k3ℓ p, where p is a prime and k,ℓ≥0 are integers, by classifying the extremal sequences of length 4p−4 modulo p2 when n=p. For every fixed n≥2, we prove wr(n)/s(Znr)=On(r2/2r)→0 as r→∞, where s(Znr) is the classical Erdős-Ginzburg-Ziv constant, with an explicit threshold on r for strict inequality. Thus, for every fixed modulus n≥2, Sun's proposed equality fails for sufficiently large r.
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BibTeX
@misc{li2026proofs,
title = {{Proofs and disproofs of some conjectures on zero-sum problems of new types}},
author = {Yang Li and San Ling and Zhonghua Sun and Shixin Zhu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19965},
eprint = {2610.19965},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19965},
}
RIS
TY - UNPB
TI - Proofs and disproofs of some conjectures on zero-sum problems of new types
AU - Yang Li
AU - San Ling
AU - Zhonghua Sun
AU - Shixin Zhu
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.19965
UR - https://arxiv.org/abs/2610.19965
AB - For integers r≥1 and n≥2, let s_r(n) be the least N such that every sequence of N vectors in Z^r, each nonzero modulo n, contains n terms whose sum is zero modulo n but nonzero modulo n^2. Let w_r(n) be the corresponding least N when any nonempty subsequence is allowed. These constants were introduced as zero-sum problems of new types by Z.-W. Sun. We prove s_r(n)≥(2^r+1−2)(n−1)+1 by constructing an explicit sequence from a lattice polytope, \emph{disproving} his conjecture s_2(n)=4n+1 for n≥4. We obtain w_r(2)=2r+1 by classifying all sequences avoiding such nonempty subsequences modulo $4$. We also \emph{prove} Sun's conjecture w_2(n)=4n−3 for n=2^k3^ℓ p, where p is a prime and k,ℓ≥0 are integers, by classifying the extremal sequences of length 4p−4 modulo p^2 when n=p. For every fixed n≥2, we prove w_r(n)/s(Z_n^r)=O_n(r^2/2^r)→0 as r→∞, where s(Z_n^r) is the classical Erdős-Ginzburg-Ziv constant, with an explicit threshold on r for strict inequality. Thus, for every fixed modulus n≥2, Sun's proposed equality fails for sufficiently large r.
ER -