Unsolvable equation

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samiul_samin
Posts:1007
Joined:Sat Dec 09, 2017 1:32 pm
Unsolvable equation

Unread post by samiul_samin » Sat Feb 24, 2018 2:46 pm

If $a$ and $b$ both are positive integer,then prove that the given equation has no solution:

$\dfrac {1}{a^2}+\dfrac {1}{ab}+\dfrac {1}{b^2}=1$

aritra barua
Posts:57
Joined:Sun Dec 11, 2016 2:01 pm

Re: Unsolvable equation

Unread post by aritra barua » Tue Feb 27, 2018 5:58 pm

Multiply the whole equation by $a^2b^2$.The equation turns to $a^2+ab+b^2$=$a^2b^2$.Add $ab$ to both sides and so $(a+b)^2-(ab)^2=ab$.Observe that $a+b$>$ab$.Notice that $(ab+1)^2-(ab)^2$=$2ab+1$>$ab$.The rest is trivial.

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