Recently I came up with a positive solution $((x,y)\neq (\pm 1;0))$ to this diophantine equation $$ x^2-\left(w^2(2^{n-2}p)^2+2^n(2^{n-2}p)\right)y^2=1,\qquad n\geq 2, $$ where all variables are in $ \mathbb{Z}^*$

Is it a known case in terms of the theory? As one can read, for instance, in the notes

• Michel Waldschmidt, Pell’s equation, author pdf.

In fact the solutions are rather elementary ideas.

Thank you.