How to show that the equation $3x^2-2y^2=1$ has infinitely many integer solutions such that $3|x$ ? ( If this can be shown then solutions of $12x^2-8y^2=4$ give infinitely many powerful numbers differing by $4$ )
If $(x,y)$ is a solution, then so is $(5x+4y, 6x+5y)$. Starting with $(1,1)$, you get a sequence of solutions. What is $x$ mod $3$ for these?
Put another way, $$ 27 \cdot 3^2 - 2 \cdot 11^2 = 1. $$ Given one $(x,y)$ pair with $$ 27 x^2 - 2 y^2 = k, $$ we get the next pair, of infinitely many, from $$ (485x + 132 y, 1782x+485 y). $$
With your $k=1$ you need start with the single seed pair $(3,11)$ to get all $(x,y)$ pairs with $x,y>0.$
For composite $k,$ you may need two or more seeds, but a small finite number in any case.