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A Short Proof of a Conjecture Regarding Quadratic Representations of Practical Numbers
Abstract
A positive integer n is called a practical number if every positive integer less than or equal to n can be expressed as a sum of distinct positive divisors of n. In this paper, we study an open conjecture proposed by Wang and Sun concerning the quadratic representations of practical numbers. Specifically, we provide a short proof of the second part of the conjecture, demonstrating that for any positive integers b and c with 2 nmid b and 2 | c, there exists an integer n satisfying 1 < n ≤ max{b, c} such that n2 + bn + c is a practical number. Combined with the work of Somu, Li, and Kukla, this completely settles the conjecture of Wang and Sun.
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BibTeX
@misc{li2026short,
title = {{A Short Proof of a Conjecture Regarding Quadratic Representations of Practical Numbers}},
author = {Ting Hon Stanford Li},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.25591},
eprint = {2608.25591},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.25591},
}
RIS
TY - UNPB
TI - A Short Proof of a Conjecture Regarding Quadratic Representations of Practical Numbers
AU - Ting Hon Stanford Li
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2608.25591
UR - https://arxiv.org/abs/2608.25591
AB - A positive integer n is called a practical number if every positive integer less than or equal to n can be expressed as a sum of distinct positive divisors of n. In this paper, we study an open conjecture proposed by Wang and Sun concerning the quadratic representations of practical numbers. Specifically, we provide a short proof of the second part of the conjecture, demonstrating that for any positive integers b and c with 2 nmid b and 2 | c, there exists an integer n satisfying 1 < n ≤ max{b, c} such that n^2 + bn + c is a practical number. Combined with the work of Somu, Li, and Kukla, this completely settles the conjecture of Wang and Sun.
ER -