Problem - 1431
Let $M$ be the product of any four consecutive positive integers. Prove $M+1$ must be a perfect square.
Let $n$ be the smallest of these four consecutive integers, then
\begin{align}
&n(n+1)(n+2)(n+3) +1 \\
&= (n(n+3))((n+1)(n+2))+1\\
&= (n^2+3n)(n^2+3n+2)+1\\
&= (n^2+3n)((n^2+3n)+2)+1\\
&= (n^2+3n)^2 + 2(n^2+3n)+1\\
&=(n^2+3n+1)^2
\end{align}