Let x, y be relatively prime integers.
Prove that x^(phi(y)) + y^(phi(x)) is congruent to 1 mod(xy)
(phi is Euler's function)
Let's multiply :
Since , we can factorise :
(2), where and hence is an integer.
Substituting (2) in (1) :
Since the last bracket is an integer, we can say that
(I tried to explain as much as possible, so excuse me if it's too long...)