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4xy - x y ≠ Perfect Square (Posted on 2009-11-01) Difficulty: 3 of 5
Prove that there does not exist any pair (x, y) of positive integers such that: 4xy - x y is a perfect square.

No Solution Yet Submitted by K Sengupta    
Rating: 2.0000 (1 votes)

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re: No Subject | Comment 2 of 3 |
(In reply to No Subject by subhobrata)

I noticed a slight problem with your proof.
you state that for (x+y)(x+y-1)-(x-y)^2 to be a perfect
sqaure then (x+y)(x+y-1)=0 but a quick counter proof to this
is if (x+y)(x+y-1)=20 and x-y=4 then you would have
20-16=4=2^2
now of course this example does not lead to positive integer x,y solutions, but it does show that your conclusion is not always true.  Although I do agree that (x+y)(x+y-1) can never be a perfect square


  Posted by Daniel on 2009-11-02 12:33:50
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