As an example, consider an integer solution of $ x ^ 2-3y ^ 2 = 13 $.
$ y $ that satisfies this equation is

$y\_k = \frac {(4+ \sqrt {3}) (2+ \sqrt {3}) ^ k-(4- \sqrt {3}) (2- \sqrt {3}) ^ k} { 2 \sqrt {3}} $

for any integer $ k $.

By calculation

$ 3 | y\_k \Leftrightarrow k \equiv1 \mod 3 $

$ 3 ^ 2 | y\_k \Leftrightarrow k \equiv7 \mod 3 ^ 2 $

$ 3 ^ 3 | y\_k \Leftrightarrow k \equiv7 \mod 3 ^ 3 $

$ 3 ^ 4 | y\_k \Leftrightarrow k \equiv7 \mod 3 ^ 4 $

$ 3 ^ 5 | y\_k \Leftrightarrow k \equiv7+2\cdot 3^4 \mod 3 ^ 5 $

$ 3 ^ 6 | y\_k \Leftrightarrow k \equiv7+2\cdot 3^4+2\cdot 3^5 \mod 3 ^ 6 $

You can see that it has a regular structure.
In other words, a solution divisible by $3^d$ appear at intervals of $3$ powers and do not appear anywhere else.
I tried similar calculations for other Pell-type equations, and I was convinced that they had a similar structure.

Can you prove this structure in general?

We consider the Pell-type equation $ x ^ 2-py ^ 2 = N $ with a solution. For any $ d $

$ p ^ d | y\_k \Leftrightarrow k \equiv r \mod p^{d+e}. $

$ r $ should be uniquely determined as $ p $-adic number.