Generalized Pell's equation where $N$ is perfect square Are all solutions of the equation $x^2-4My^2=K^2$, multiples of $K$? I am considering $M$ not perfect square. Any tests in Python show be true, but...
My code:
for x in range (1,8000):
    for y in range (1,8000):
            if (x*x-20*y*y)==36:
                print(x,y)

Result: $
54, 12\\
966, 216$
All multiples of $6$.
Any can a example of solution coprime (except by factor 2) with K?

$$  21^2 - 20 \cdot 4^2 = 11^2  $$
$$  21^2 - 20 \cdot 2^2 = 19^2  $$
$$  61^2 - 20 \cdot 12^2 = 29^2  $$
$$  41^2 - 20 \cdot 6^2 = 31^2  $$
$$  49^2 - 20 \cdot 6^2 = 41^2  $$
$$  69^2 - 20 \cdot 8^2 = 59^2  $$
$$  101^2 - 20 \cdot 18^2 = 61^2  $$
$$  89^2 - 20 \cdot 12^2 = 71^2  $$
$$  81^2 - 20 \cdot 4^2 = 79^2  $$
$$  161^2 - 20 \cdot 30^2 = 89^2  $$
$$  141^2 - 20 \cdot 22^2 = 101^2  $$
$$  141^2 - 20 \cdot 20^2 = 109^2  $$
