perfect square problem

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Joined: Aug 4, 2011

by hemangsarkar, Sep 6, 2012, 6:36 AM

Q) let $x, y$ be two positive integers, prove that if
$\left(\frac{x^2+ y^2}{2} \right)(xy)$ is a perfect square, then $x = y$ .

my solution :
let $y = mx$ , where $m = \frac{a}{b}$ and $gcd(a,b) = 1$ .

then $\left( \frac{m^2x^2 + x^2}{2} \right)(mx^2)$ is a perfect square.

or, $(m^2 + 1) (\frac{m}{2})$ is a perfect square.

$\frac{(a^2 + b^2)a}{2b^3} = n^2$ , for some $n$ .

$a(a+b)^2 = 2b(a^2 + nb^2)$ .

so either $b$ divides $a$ , or $b$ divides $(a+b)$ .

also, $gcd(a,b) = 1$ .

so $a = b = 1$ .

$y = x$ .

note : i am not very sure of my solution. please tell me of some mistake, you see.

This post has been edited 1 time. Last edited by hemangsarkar, Sep 6, 2012, 6:37 AM

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perfect square problem

by hemangsarkar, Sep 6, 2012, 6:36 AM

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