# Prove the Product of four consecutive positive integers cannot be a perfect Square

Let N be the first of four consecutive positive integers.

Then assume that $N\left(N+1\right)\left(N+2\right)\left(N+3\right)$ is a perfect square.

Also

$N\left(N+1\right)\left(N+2\right)\left(N+3\right)=\left({N}^{2}+3\,N\right)\left({N}^{2}+3\,N+2\right)=\left({N}^{2}+3\,N+1\right)^{2}-1$

Since two squares cannot differ by one, we have shown $N\left(N+1\right)\left(N+2\right)\left(N+3\right)$ 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.