rational solutions(BOMC-2)

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Tahmid Hasan
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rational solutions(BOMC-2)

Unread post by Tahmid Hasan » Tue Mar 27, 2012 11:31 pm

Prove that the equation $x^3+y^3=9$ has infinitely many rational solutions.
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Nadim Ul Abrar
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Re: rational solutions(BOMC-2)

Unread post by Nadim Ul Abrar » Wed Mar 28, 2012 7:32 pm

Collected :
If $(x,y)$ be a rational solution to the equation $x^3+y^3=A$
then $(x',y')=(\frac{y(x^3+A)}{y^3-x^3},\frac{x(y^3+A)}{x^3-y^3})$
is also a solution .

(The source didn't gave the proof , But Putting $A=x^3+y^3$ ,
if we simplify the expression then we will get $x^3+y^3$
$(\frac{y(x^3+A)}{y^3-x^3})^3+(\frac{x(y^3+A)}{x^3-y^3})^3=x^3+y^3$)

Now we have a rational solution namely (2,1) . So we can find infinitly many solutions
$\frac{1}{0}$

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Sazid Akhter Turzo
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Re: rational solutions(BOMC-2)

Unread post by Sazid Akhter Turzo » Wed Mar 28, 2012 8:03 pm

What's the source?
Turzo

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Tahmid Hasan
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Re: rational solutions(BOMC-2)

Unread post by Tahmid Hasan » Thu Mar 29, 2012 5:11 pm

Here's the source from where i took the problem
https://www.artofproblemsolving.com/For ... hp?t=29434.
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