Let’s have the equation $a^k+\frac{k \cdot a^k}{x \cdot y^n}=m^k$ where $k≥2$ and $x≠y$ and $a, k, x, y, n, m$ positive integers greater than zero. If $x \cdot y^n = f \cdot a^g$ where $f$ and $g$ any positive integer greater than zero then the equation $ a^k + \frac{k \cdot a^k}{x \cdot y^n}=L$ has infinite solutions where L is an integer but never has the form $L=m^k$. Does anyone know what will be the easiest way to prove that the above equation is impossible for all values $k≥2$?