Multi-exponential diophantine equation

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Fm Jakaria
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Multi-exponential diophantine equation

Unread post by Fm Jakaria » Thu Jul 04, 2013 10:26 pm

Find all integer solutions $(s,u,v)$ to the diophantine equation;
\[s^3 = u^2 + 3 v^2\]
given that $s$ is odd.

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Individ
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Re: Multi-exponential diophantine equation

Unread post by Individ » Sun Feb 01, 2015 11:57 am

For the equation: $$X^2+qY^2=Z^3$$

You can write this simple solution:

$$X=(p^2+qs^2)((p^4-q^2s^4)t^3-3(p^2+qs^2)^2kt^2+3(p^4-q^2s^4)tk^2-(p^4-6qp^2s^2+q^2s^4)k^3)$$

$$Y=2ps(p^2+qs^2)((p^2+qs^2)t^3-3(p^2+qs^2)tk^2+2(p^2-qs^2)k^3)$$

$$Z=(p^2+qs^2)((p^2+qs^2)t^2-2(p^2-qs^2)tk+(p^2+qs^2)k^2)$$

$q - $The ratio is given for the problem. $p,s,t,k - $ integers asked us.

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