Congruence Classes of Supporting the Erdös-Straus Conjecture I: Tame Solutions
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arXiv
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2026
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| _version_ | 1866918518529720320 |
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| author | Xu, Xiaoping |
| author_facet | Xu, Xiaoping |
| contents | 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 one only needs to prove the conjecture for any prime number $n$ such that $n\equiv 1\;(\mbox{mod}\;24)$. 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. Computer calculation shows that there are only nine wild primes among the 7185 primes of the form $24m+1$ with $m\leq 30000$. In this paper, we derive the tame solutions of the above equation for the integers of the form $24m+1$ with $m$ parameterized by certain congruence classes. They cover the solvability of all the 586 tame primes among the 591 primes of the form $24m+1$ with $m\leq 2000$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_23601 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Congruence Classes of Supporting the Erdös-Straus Conjecture I: Tame Solutions Xu, Xiaoping Number Theory Primary 11D68: Secondary 11D85, 11A67, 11B75 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 one only needs to prove the conjecture for any prime number $n$ such that $n\equiv 1\;(\mbox{mod}\;24)$. 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. Computer calculation shows that there are only nine wild primes among the 7185 primes of the form $24m+1$ with $m\leq 30000$. In this paper, we derive the tame solutions of the above equation for the integers of the form $24m+1$ with $m$ parameterized by certain congruence classes. They cover the solvability of all the 586 tame primes among the 591 primes of the form $24m+1$ with $m\leq 2000$. |
| title | Congruence Classes of Supporting the Erdös-Straus Conjecture I: Tame Solutions |
| topic | Number Theory Primary 11D68: Secondary 11D85, 11A67, 11B75 |
| url | https://arxiv.org/abs/2605.23601 |