Nontrivial Solutions

**diddledabble** · August 5th 2009, 08:52 AM

Show that $x^2+y^2=4xy$ has no notrivial solutions. I am not sure where to begin. Is it like a Fermat Pell equation?

**Bruno J.** · August 5th 2009, 10:53 AM

If it has a nontrivial solution, it clearly has one with $x,y$ coprime. But, for example, $y^2=4xy-x^2$ implies that $x$ divides $y^2$ ... can you finish?

**diddledabble** · August 5th 2009, 11:23 AM

No, you lost me. Should I rewrite as $x^2+x^2-4xy=4xy$ ? Then $2x^2=8xy$

**Gamma** · August 5th 2009, 12:16 PM

No, those two equations are not the same.

What he is saying get the original equation in the form he has it (subtract $x^2$ from both sides. Then you see the RHS you can factor out an x to see that x must divide $y^2$ . But if $x|y^2$ , then you see that x and y must share at least 1 prime factor, or else x is $\pm 1$ or y is 0. y=0 yields the trivial solution. $x=\pm 1$ has no integer solutions, use the quadratic formula to see why (or rational root test if you want). So they must share a prime factor making them not relatively prime.

Very nice proof bruno.

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