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April 9th, 2014, 12:47 AM   #1
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a perfect square problem

Prove or disprove the claim that there are integers m, n such m^2 + mn + n^2 is a perfect square.

My attempt:

You can't factorize m^2 + mn + n^2 into two equal factors. Therefore, the claim that there are integers m, n such that m^2 + mn + n^2 is a perfect square is false.

Am I correct? Is there a better way to prove this? thanks
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April 9th, 2014, 12:59 AM   #2
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What about $m=n=0$? Or $(m,n)=(0,1),\,(1,0)$? Or $m=-n$?
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Last edited by MarkFL; April 9th, 2014 at 01:02 AM.
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April 9th, 2014, 01:43 AM   #3
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The fact that an expression can’t be factorized does not imply that it can’t be a perfect square. For example the expression $2x^2+2x+1$ can’t be factorized (technically we say that the polynomial is irreducible over $\mathbb Z$) but it is a perfect square for $x=3$.
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April 9th, 2014, 07:18 AM   #4
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Quote:
Originally Posted by Olinguito View Post
The fact that an expression can’t be factorized does not imply that it can’t be a perfect square. For example the expression $2x^2+2x+1$ can’t be factorized (technically we say that the polynomial is irreducible over $\mathbb Z$) but it is a perfect square for $x=3$.
Yes, exactly. In slightly more technical terms: factorization over $\mathbb{Z}[x]$ is not necessary for factorization over $\mathbb{Z}.$
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