<s>[INST] A caboodle of Pell's equation in one? $x^2+y^2-5xy+5=0$ I saw this twitter post that reads:

Find all the pairs of positive integers $(x,y)$ satisfying $$  x^2 + y^2 - 5xy + 5 = 0 . $$

I don't know how to tackle this and I ended up summoning WolframAlpha which shows that there are infinitely many solutions. What's interesting is that there are (at least?) 12 general forms that looks like they all resemble the solutions to Pell's equation.
For example, the first general solution presented reads:

$$ \begin{array} { r c l }
 x &=& 
\dfrac1{42} \Big [ 21 \left(55 - 12\sqrt{21} \right)^n 
- \sqrt{21} \left(55 - 12\sqrt{21} \right)^n 
+ 21 \left(55 + 12\sqrt{21} \right)^n 
+ \sqrt{21} \left(55 + 12\sqrt{21} \right)^n \Big ] \\ \phantom0 \\
 y &=& 
\dfrac1{42} \Big [ 63 \left(55 - 12\sqrt{21} \right)^n 
- 13 \sqrt{21} \left(55 - 12\sqrt{21} \right)^n 
+ 63 \left(55 + 12\sqrt{21} \right)^n 
+ 13 \sqrt{21} \left(55 + 12\sqrt{21} \right)^n \Big] \end{array} $$ for $n = 0,1,2,3\ldots $.

Plugging $n=0,1,2,3$ gives the first few solutions
$$(x,y) = (1,3), (67,321), (7369,35307), (810523,3883449) 
$$
I tried converting these solutions to the form of $X^2 - DY^2 = A$ or something similar to Pell's equation but I got nothing.
Is it a coincidence that this innocuous-looking quadratic Diophantine equation is actually a caboodle of Pell's equation in disguise? If so, how can we derive them all?
 [/INST] It's $(5y-2x)^2-21y^2=-20$, which is a Pell type equation.

I got that by completing the square:
$x^2-5xy+y^2=-5\implies \left(x-\frac52y\right)^2-\frac{21}4y^2=-5\implies (5y-2x)^2-21y^2=-20$.
So it's $X^2-21Y^2=-20$, with $X=5y-2x$ and $Y=y$.
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