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Solution of MMM#38

November 2nd, 2009

MMM #38:

Prove or disprove: The product of any four consecutive integers is always one less than a perfect square.
Don’t assume the integers are all positive.

Any four consecutive integers can be written as $n-2,n-1,n,n+1$ . The product of these numbers is

$\begin{align*}(n-1)n(n+1)(n-2)&=(n^2-n)(n^2-n-2)&\\&=\left[(n^2-n-1)+1\right]\left[(n^2-n-1)-1\right]\\&=(n^2-n-1)^2-1.\end{align*}$

Therefore the statement always true.

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