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 April 9th, 2014, 12:47 AM #1 Senior Member     Joined: Oct 2013 From: Far far away Posts: 431 Thanks: 18 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
 April 9th, 2014, 12:59 AM #2 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs What about $m=n=0$? Or $(m,n)=(0,1),\,(1,0)$? Or $m=-n$? Thanks from shunya Last edited by MarkFL; April 9th, 2014 at 01:02 AM.
 April 9th, 2014, 01:43 AM #3 Senior Member     Joined: Apr 2014 From: Greater London, England, UK Posts: 320 Thanks: 156 Math Focus: Abstract algebra 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 $x=3$. Thanks from CRGreathouse
April 9th, 2014, 07:18 AM   #4
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Quote:
 Originally Posted by Olinguito 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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