Question

If a,b,c,d are four consecutive integers, prove that √abcd+1 is a rational number.

Answer 1

Let the four consecutive integers a,b,c,d be $n,~n+1,~n+2,~n+3$ and consider 
$n(n+1)(n+2)(n+3)+1$

As $1+2~=~0+3$, $n(n+1)(n+2)(n+3)+1 = \underbrace{n(n+3)}\underbrace{(n+1)(n+2)}+1~=~(n^2+3n)(n^2+3n+2)+1$

Letting $n^2+3n=k$ :
$k(k+2)+1~=~k^2+2k+1~=~(k+1)^2$

The squareroot of which is an integer.