Recently i came up with a positive solution $((x,y)\neq (\pm 1;0))$ to this diophantine equation

$$x^2-(w^2(2^{n-2}p)^2+2^n(2^{n-2}p))y^2=1$$

$n\ge 2$ where variables are non zero relative integers.

Is it a known case in terms of the theory? As one can read in this file

https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/BamakoPell2010.pdf

In fact the solutions are rather elementary ideas.

Thank you.