A note on the Erdős–Straus conjecture

In this note, we consider the Erdős–Straus Diophantine equation $$\begin{aligned} \frac{c}{n}=\frac{1}{x} + \frac{1}{y} + \frac{1}{z}, \end{aligned}$$ where n and c are positive integers with \(\gcd (n, c) = 1\). We provide a formula for the number f(n, c) of all positive integral solutions (x, y, z) of the equation. In 1948, Erdős and Straus conjectured that \(f(n,4) \ge 1,\) for all integers \(n \ge 2\). Here, we solve the conjecture for a special case of n.