Prove that product of four consecutive positive integers increased by 1 is a perfect square.

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Solution

Let the consecutive positive integers be $n, n + 1, n + 2 a n d n + 3$
Consider the expression,
$N = n (n + 1) (n + 2) (n + 3) + 1$
$= (n^{2} + 3 n) (n^{2} + 3 n + 2) + 1$
$= (n^{2} + 3 n)^{2} + 2 (n^{2} + 3 n) + 1$
$= {[(n^{2} + 3 n) + 1]}^{2}$
$= (n^{2} + 3 n + 1)^{2}$

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