I need to prove the following, but I am not able to do it, this is not homework, nor something related to research, this question came in my exam and so I would want to have a solution which is possible in minutes(Exam)
Q. If $n = 1 + m$, where $m$ is the product of four consecutive positive integers, prove that $n$ is a perfect square.
Now since $m = p(p+1)(p+2)(p+3)$;
$p = 0, n = 1$ Perfect Square
$p = 1, n = 25$ Perfect Square
$p = 2, n = 121$ Perfect Square
Is there any way to prove the above without induction. My approach was to expand $m = p(p+1)(p+2)(p+3)$ into a 4 degree equation, and then try making $n = m + 1$ as a perfect square, but I wasn't able to do it. Any idea if it is possible?