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

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    2016
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The discussion centers on proving that the product of four consecutive positive integers cannot be a perfect square. Participants engage with mathematical reasoning and provide proofs to support this claim. Two members, kaliprasad and lfdahl, successfully present correct solutions to the problem. The thread emphasizes the importance of understanding integer properties and their implications in number theory. Overall, the conversation highlights a significant mathematical concept regarding the nature of products of consecutive integers.
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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!
 
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Congratulations to the following members for their correct solution::)

1. kaliprasad
2. lfdahl

Here's the proposed solution:
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.
 
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