What's important here is that we are trying to solve for two natural integers.
If we assume that is odd then is even and is odd so can't hold : can't be odd.
Now let's assume that is even. As 2 divides , 4 divides . As is odd, 4 doesn't divide hence from we get that 4 must divide : for some . We can now write as which is an equation in terms of .
Using the discriminant we get that the solutions of this equation are and . We know that is an integer so we are looking for the values of such that at least one of the two solutions is an integer. When does this happen ?