Congruence Classes of Supporting the Erd\"{o}s-Straus Conjecture I: Tame Solutions

In 1948, Erd\"{o}s 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 $m\leq 2000$.