Proving the Non-Perfect Square Property of 4 Consecutive Positive Integers

[anemone]

Here is this week's POTW:

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Prove that the product of 4 consecutive positive integers is never a perfect square.

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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!

 

[anemone]

Congratulations to the following members for their correct solution::)

1. kaliprasad
2. lfdahl

Here's the proposed solution:

Spoiler

Let $n,\,n+1,\,n+2$, and $n+3$ be the four consecutive positive integers.

Observe that

$n(n+1)(n+2)(n+3)=(n^2+3n)(n^2+3n+2)=k(k+2)$, where $k=n^2+3n$, but $k^2+2k$ is never a square since

$k^2<k^2+2k<(k+1)^2$

Therefore we can conclude by now that the product of 4 consecutive positive integers is never a perfect square.