In this note, we consider the Erd?s–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, Erd?s 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.