Logo
All Random Solved Random Open
OPEN
Is it true that there are no solutions to \[n! = x^k\pm y^k\] with $x,y,n\in \mathbb{N}$, with $xy>1$ and $k>2$?
Erdős and Obláth [ErOb37] proved this is true when $(x,y)=1$ and $k\neq 4$. Pollack and Shapiro [PoSh73] proved this is true when $(x,y)=1$ and $k=4$. The known methods break down without the condition $(x,y)=1$.
Additional thanks to: Zachary Chase