More prime number mumbo jumbo

Let be the kth prime number. Show that cannot be the perfect square of an integer.

Like most "proof this cannot be", we assume it's true, and then show that it's not possible. So, we rewrite this as:



What's that? A difference of squares?? Factor the right side:


Now we have something we can work with. First, let's replace with 2, since we know 2 is the first prime:



From this, we can deduce that (x+1)(x-1) is divisible by 2, and is thus even. The only way this is possible is if x is odd. (Odd + 1 = Even, but Even + 1 = Odd) This also means that both (x+1) and (x-1) are even. Now, this is where we can find our contradiction. Let's divide both sides by 2. What we are left with on the left side is . We know that 2 is the only even prime number, so all of these numbers must be odd. But, since both (x+1) and (x-1) are even, we have another factor of two on the right side. And since is all odd numbers, it isn't divisible by 2. What does this mean? It means we've made a mistake, and that mistake was our original assumption, that is the square of an integer. Therefore, through assuming it's true and pointing out a contradiction, we've proved that it isn't possible.