<s>[INST] Solutions of the Diophantine equation $x^2(x^2+10)=3y^2(y^2+10)$ I am looking for the solutions of the Diophantine equation 
$$x^2(x^2+10)=3y^2(y^2+10).$$
Is there any solution of this equation except when $(x,y)=(0,0)$?
Or
Any computer programme such as MAGMA could solve this problem?
Thank you very much.
 [/INST] If $x$ and $y$ are not both $0$, let $3^k$ be the highest power of $3$ that divides both $x$ and $y$. Let $x=3^ks$ and $y=3^k t$. Then $3$ cannot be a common divisor of $s$ and $t$.
Substitute and cancel. We get $s^2(x^2+10)=3t^2(y^2+10)$. Since $3$ cannot divide $x^2+10$, it must divide $s$. Say $s=3u$. Then $9u^2(x^2+10)=3t^2(y^2+10)$, and therefore $3u^2(x^2+10)=t^2(y^2+10)$. But then $3$ divides $t$, contradicting the fact that $3$ is not a common divisor of $s$ and $t$.   
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