Okay, I think I came up with a proof.

Suppose , then it means that .

This means that:

Or, written in an other way:

Now, for , since is prime, and are coprimes if and only if .

And that's true because, we supposed that , so if we would have , but , so that's absurd.

This imply that in the previous equation, the right hand side is an integer , but the denominator does not divide ,

which means that the denominator must divide the rest of the numerator, but this imply that , which is absurd because we supposed

to be the biggest exponent such that .

Thus