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arXiv · 2608.25591

A Short Proof of a Conjecture Regarding Quadratic Representations of Practical Numbers

Ting Hon Stanford Li
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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 $2 \mid c$, there exists an integer $n$ satisfying $1 < n \le \max\{b, c\}$ 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.

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Ting Hon Stanford Li. 2026-08-26. A Short Proof of a Conjecture Regarding Quadratic Representations of Practical Numbers. https://arxiv.org/abs/2608.25591

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