AI 中文总结
本文针对Wang和Sun提出的实用数二次表示的猜想，通过证明特定二次式取值为实用数的结论，结合他人工作完整解决了该猜想。
AI 中文摘要
正整数n被称为实用数，当且仅当每个不超过n的正整数都可表示为n的不同正除数之和。本文研究了Wang和Sun提出的关于实用数二次表示的公开猜想，具体而言，我们给出了该猜想第二部分的简短证明，证明对于任意满足2不整除b且2整除c的正整数b和c，存在整数n满足1 < n ≤ max{b, c}，使得n² + bn + c为实用数。结合Somu、Li和Kukla的工作，这一结果完全解决了Wang和Sun的猜想。
英文摘要
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.
CommentsFixed a logical gap in Lemma 4 by replacing the modular Vieta's formulas argument with an explicit parity-based factorization step. Minor expositional improvements throughout
