支持 Erdös-Straus 猜想 的同余类 II：野解

中文摘要 AI 辅助

1948 年，Erdös 和 Straus 提出了一个猜想：对于任意正整数 $n>2$，存在正整数 $n_1,n_2$ 和 $n_3$ 使得 \begin{equation}\frac{4}{n}=\frac{1}{n_1}+\frac{1}{n_2}+\frac{1}{n_3},\nonumber\end{equation} 该猜想至今仍未解决。已知若对任意素数 $n\equiv 1\\;(\mbox{mod}\\;24)$ 证明该猜想成立，则猜想成立。若 $n=24m+1$ 且 $n_1\leq n_2,n_3$，则 $n_1=6m+k$ 其中 $1\leq k\leq 12m$。上述方程的解 $(n_1,n_2,n_3)$ 称为驯化解，若 $n_2$ 和 $n_3$ 是 $(6m+k)(24m+1)$ 的因子。我们称 $n=24m+1$ 为野的，若它没有驯化解。基于我们先前在 arXiv 上关于驯化解的工作信息，Howerton 发现形式为 $n=24m+1\leq 2.4\times 10^{11}$ 的野素数仅有十四个。在本文中，我们推导出上述方程的三十四个野解族，其中包含这十四个野素数的可解性。连同我们早先的驯化多项式解，数值测试表明它们覆盖了所有形式为 $24m+1$ 的素数。

英文摘要

In 1948, Erdös and Straus formulated a conjecture : for any positive integer $n>2$, there exist positive integers $n_1,n_2$ and $n_3$ such that \begin{equation}\frac{4}{n}=\frac{1}{n_1}+\frac{1}{n_2}+\frac{1}{n_3},\nonumber\end{equation} which is still open. It is known that the conjecture holds if one can prove it for any prime $n\equiv 1\;(\mbox{mod}\;24)$. If $n=24m+1$ and $n_1\leq n_2,n_3$, then $n_1=6m+k$ with $1\leq k\leq 12m$. A solution $(n_1,n_2,n_3)$ of the above equation is called a {\it tame solution} if $n_2$ and $n_3$ are factors of $(6m+k)(24m+1)$. We call $n=24m+1$ {\it wild} if it does not have any tame solution. Based on the information in our earlier work on tame solutions posed in arXiv, Howerton found that there are only fourteen wild primes of the form $n=24m+1\leq 2.4\times 10^{11}$. In this paper, we derive thirty-four families of wild solutions of the above equation, which contain the solvability of the fourteen wild primes. Together with our earlier tame polynomial solutions, numeric test shows that they cover all the primes of the form $24m+1$.

发表机构

• HLM, Institute of Mathematics, Academy of Mathematics & System Sciences Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
• School of Mathematics, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）

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