<s>[INST] Solving a Complex Diophantine Equation with One Known I am struggling to find a way to implement Euclid's Algorithm in order to solve this diophantine equation. The $N$ will be known and the set of solutions I wish to find will be a set of decreasing $X$ solutions that satisfy $$x^2 - 4  y^2 = n.$$
When simplifying I have concluded that the solution could be as follows...
$$x^2 - 4 y^2 = (x - 2y)  (x + 2y)$$ when $n = 12$,
$$12 = (x - 2y) (x + 2y)$$
would it be possible to use Euclid's Algorithm to solve for the GCD in decreasing order?
The poposed solutions are a set of $n=12$, $x = 4$ and $y = 1$.
 [/INST] Say $12=(x+2y)(x-2y)=ab$, where $x$ and $y$ are positive integers,
with $x+2y=a, $ and $x-2y=b$.  Then $a>b$, and $2x=a+b$.
That means $a+b$ must be even, so $a$ and $b$ are both odd or both even.
But $a$ and $b$ cannot be both odd, since $ab=12$ is even.
Therefore, $a$ and $b$ are both multiples of $2$.
Since $12=2\times2\times3=ab$, it follows that $a=2\times3$ and $b=2$,
so $x=(a+b)/2=4$ and $y=1$.
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