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Counterexamples to the Minimum Period Conjecture for Restricted Partition Functions
Abstract
For a finite sequence of positive integers a=(a1,…,an), the restricted partition function qa(k) denote the number of nonnegative integer solutions to the equation a1x1+a2x2+⋯ +anxn=k. It is proved to be a quasi-polynomial of degree n−1. Write qa(k)=∑j=0n−1cj(k)kj with periodic coefficient functions cj, and set bm=#{i:m| ai}. In 2008, Beck, Sam, and Woods conjectured that the minimum period of cj(k) is lcm{m:bm>j}. In this paper, we derive an exact root-of-unity formula for every coefficient function cj(k). The formula proves the conjectured divisibility upper bound, but it also reveals a lower bound for the period of cj(k). Both divisibility bounds are sharp. This leads us to construct a family of counterexamples to this conjecture.
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BibTeX
@misc{liu2026counterexamples,
title = {{Counterexamples to the Minimum Period Conjecture for Restricted Partition Functions}},
author = {Feihu Liu and Jinlong Tang and Guoce Xin and Chen Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.02085},
eprint = {2608.02085},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.02085},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Counterexamples to the Minimum Period Conjecture for Restricted Partition Functions
AU - Feihu Liu
AU - Jinlong Tang
AU - Guoce Xin
AU - Chen Zhang
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2608.02085
UR - https://arxiv.org/abs/2608.02085
AB - For a finite sequence of positive integers a=(a_1,…,a_n), the restricted partition function q_a(k) denote the number of nonnegative integer solutions to the equation a_1x_1+a_2x_2+⋯ +a_nx_n=k. It is proved to be a quasi-polynomial of degree n−1. Write q_a(k)=∑_j=0^n−1c_j(k)k^j with periodic coefficient functions c_j, and set b_m=#{i:m| a_i}. In 2008, Beck, Sam, and Woods conjectured that the minimum period of c_j(k) is lcm{m:b_m>j}. In this paper, we derive an exact root-of-unity formula for every coefficient function c_j(k). The formula proves the conjectured divisibility upper bound, but it also reveals a lower bound for the period of c_j(k). Both divisibility bounds are sharp. This leads us to construct a family of counterexamples to this conjecture.
ER -