Overview
The Erdős-Straus Conjecture, first formulated in 1948, posits that for any positive integer $n \geq 2$, there exist positive integers $n_1, n_2,$ and $n_3$ satisfying the equation $\frac{4}{n}=\frac{1}{n_1}+\frac{1}{n_2}+\frac{1}{n_3}$. This conjecture remains unproven. Previous work has established that the conjecture's validity hinges on proving it for prime numbers $n \equiv 1 \pmod{24}$.
Research Context
The conjecture's solvability for primes of the form $n=24m+1$ is a particular focus. For such $n$, and under the assumption that $n_1 \leq n_2, n_3$, $n_1$ can be expressed as $6m+k$, where $1 \leq k \leq 12m$. Within this framework, a solution $(n_1, n_2, n_3)$ is classified as a "tame solution" if $n_2$ and $n_3$ are factors of the product $(6m+k)(24m+1)$. Conversely, a prime $n=24m+1$ is termed "wild" if it does not possess any tame solution.
Prior research, building on information from an earlier work on tame solutions published on arXiv, identified a specific set of wild primes. Howerton's analysis revealed that there are precisely fourteen wild primes of the form $n=24m+1$ within the range $n \leq 2.4 \times 10^{11}$.
Approach
This paper focuses on the derivation of "wild solutions" for the Erdős-Straus conjecture, particularly for primes of the form $n=24m+1$. The methodology involves deriving specific families of solutions that address these previously identified wild primes.
Findings
- The study successfully derived thirty-four distinct families of wild solutions for the equation $\frac{4}{n}=\frac{1}{n_1}+\frac{1}{n_2}+\frac{1}{n_3}$.
- These thirty-four families of wild solutions contain the solvability for all fourteen wild primes previously identified by Howerton, which are of the form $n=24m+1$ and are less than or equal to $2.4 \times 10^{11}$.
- When these newly derived wild solutions are combined with previously established tame polynomial solutions, numeric tests indicate that they collectively cover all primes of the form $24m+1$. This suggests that for any prime $n=24m+1$, a solution to the Erdős-Straus Conjecture can be found within these categories of solutions.