My professor gave us a worksheet with diophantine equations including this one he claims that it is one of the easier ones but that it has a unique solution.
Find all integer solutions to the equation $$x^2 − x = y^5 − y$$.
Also my professor will give extra credit for all prime number solutions (x and y).
Rearranging the equation gives $$(x^2-y^2)-(x-y)=0$$ Now note that $x^2-y^2=(x-y)(x+y)$.
I'll let you take it from here.