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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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 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 $\mathbb{Z}.$ Tags perfect, problem, square Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post harrypham Number Theory 8 July 19th, 2012 03:12 PM greg1313 Number Theory 5 November 28th, 2011 06:16 AM elim Number Theory 6 September 10th, 2011 11:00 PM PRO Number Theory 6 August 3rd, 2011 05:38 PM calligraphy Number Theory 4 February 10th, 2011 05:34 AM

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