I’ve noticed that taking two prime numbers $x$ and $y$ given $y>x$, and computing $(x^y+y)-(y^x+x)$ gives a number that’s always divisible by $x$ and $y$, provided that $x$ and $y$ are prime.
For example, $$(3^5+5)-(5^3+3)=120,$$ and $$120=(3 \times 5) \times 8.$$
Does anyone have an idea why this could be true, and could one rigorously prove that this is true for all prime numbers?
