Let N be the first of four consecutive positive integers.

Then assume that is a perfect square.

Also

Since two squares cannot differ by one, we have shown is NOT a perfect square!

Note that another way of looking at this problem is that we proved that the product of four consecutive positive integers increased by 1 is a perfect square.

If you liked the logic associated with this mathematical proof, then you would probably also like chess which involves the same type of reasoning and thinking.

Closely related to the above proof is to show that the product of 2 consecutive positive integers cannot be the product of 4 consecutive positive integers.