<s>[INST] Finding all pairs $(a,b)$ of positive integers such that $a^2+nab+b^2$ is a perfect square. When $n=2$, the question is trivial. Is there a general method to find all such pairs for $n\ge{3}$ and $n\in{\mathbb{N}}$?
 [/INST] Generally speaking, this equation has a lot of formulas for the solution. Because it is symmetrical.
Write the formula can someone come in handy. the equation:
$$Y^2+aXY+X^2=Z^2$$
Has a solution:
$$X=as^2-2ps$$
$$Y=p^2-s^2$$
$$Z=p^2-aps+s^2$$
more:
$$X=(4a+3a^2)s^2-2(2+a)ps-p^2$$
$$Y=(a^3-8a-8)s^2+2(a^2-2)ps+ap^2$$
$$Z=(2a^3+a^2-8a-8)s^2+2(a^2-2)ps-p^2$$
more:
$$X=(a+4)p^2-2ps$$
$$Y=3p^2-4ps+s^2$$
$$Z=(2a+5)p^2-(a+4)ps+s^2$$
more:
$$X=8s^2-4ps$$
$$Y=p^2-(4-2a)ps+a(a-4)s^2$$
$$Z=-p^2+4ps+(a^2-8)s^2$$
In the equation:  $$X^2+aXY+bY^2=Z^2$$ there is always a solution and one of them is quite simple.
$$X=s^2-bp^2$$
$$Y=ap^2+2ps$$
$$Z=bp^2+aps+s^2$$
$p,s$ - integers asked us.
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