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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 $m=n$?
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 $x=3$.

April 9th, 2014, 07:18 AM  #4  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Quote:
 

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