Pretty Diophantine Equation

For discussing Olympiad Level Number Theory problems
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Kazi_Zareer
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Pretty Diophantine Equation

Unread post by Kazi_Zareer » Tue Dec 20, 2016 11:21 am

Find all pairs of integers $ (x,y)$, such that


\[ x^2 - 2009y + 2y^2 = 0

\]
We cannot solve our problems with the same thinking we used when we create them.

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ahmedittihad
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Re: Pretty Diophantine Equation

Unread post by ahmedittihad » Fri Jan 13, 2017 6:21 am

The only solution is $(21,28)$.
Frankly, my dear, I don't give a damn.

dshasan
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Re: Pretty Diophantine Equation

Unread post by dshasan » Mon Feb 27, 2017 7:14 pm

Note that the equation can be rewritten as $m_2^2 + 8n_3^2 = 41n_3$, where $x = 7\times 7\times 2\times m_2$ and $y = 7\times 7\times 2\times 2\times n_3$, which gives the only solution when $(m_2, n_3) = (6,4) \Rightarrow (x,y) = (588,784)$ :D


Last bumped by dshasan on Mon Feb 27, 2017 7:14 pm.
The study of mathematics, like the Nile, begins in minuteness but ends in magnificence.

- Charles Caleb Colton

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