prove that if a+ b/a - 1/b is an integer then it is a perfect square for integers a, b.
By setting a=-1, b=1 you can see the statement does not hold.
But let's try to fix it: "if a+ b/a - 1/b is an integer then it is a perfect square for positive integers a, b."
So let be an integer for positive integers . Then for some integer . We get .
We see this implies .
If we immediately see that our statement holds.
We'll finish the proof by showing that we cannot have . If it is, then implies . After dividing by we get . This means that divides , so we cannot have , which is a contradiction.