# Solution to the Pell Equation and Pell like Equation

I have read through a lot of similar posts so I am not trying to re ask a question but just seeking some clarity.

I am looking at the Pell and Pell like equations:

$$x^2-2y^2=1$$ and $$x^2-2y^2=k$$ where $$k=6n+1$$ for $$n \in \mathbb N$$. I am just curious to know if all solutions have either $$x$$ or $$y \equiv 0 \mod 3$$ $$\quad$$ or if I am thinking about this completely wrong.

So for a problem that I am looking at the smallest solution would be $$(3,2)$$ because $$(1,0)$$ would not make sense. So when I used the seed it gave solutions with either $$x,y$$ divisible by 3 but I am sure my thinking is flawed.

Taken mod $$3$$, any equation of the form $$x^2-2y^2=6n+1$$ becomes $$x^2+y^2\equiv1$$ mod $$3$$. If $$xy\not\equiv0$$ mod $$3$$, then $$x^2\equiv y^2\equiv1$$ mod $$3$$, in which case $$x^2+y^2\equiv2\not\equiv1$$ mod $$3$$. So we must have $$xy\equiv0$$ mod $$3$$, which is to say $$3\mid x$$ or $$3\mid y$$ (since $$3\mid xy$$ and $$3$$ is prime).