A diophantine equation inspired in a conjecture due to Gica and Luca, example of a large Mersenne exponent

Question

In this post I consider the equation $$k\cdot x=y^2+z^2(x^2-2)-2\tag{1}$$ over odd integers $y\geq 1$ and $z\geq 1$, and over integers $k\geq 1$ and very large Mersenne exponents $x$ such that $x^2-2$ is a prime number.

Previous Diophantine equation $(1)$ is consequence of Fermat's little theorem applied to the diophantine equation studied by professors Alexandru Gica and Florian Luca in Conjecture 4 of [1]. Mersenne exponents is the sequence A000043 from The On-Line Encyclopedia of Integer Sequences (I add that also Wikipedia has the article Mersenne prime).

I wondered about this problem that I've stated after I've realized that the $x's$ of the solutions $(x,y,z)$ of professors in [1], in the context of their Conjecture 4, are Mersenne exponents. I'm asking this post as curiosity, similar to the post that I've edited for $x=61$ in Mathematics Stack Exchange post [2] with identificator 4479581.

Question. I would like to know if it is possible to do some work in order to find or characterize all solutions of the corresponding equation $(1)$, over $k\geq 1$ integer, and over odd integers $y\geq 1$ and $z\geq 1$ of the diophantine equation $$25964951\cdot k=y^2+674178680432399\cdot z^2-2.\tag{2}$$ (This is the specialization of $(1)$ for the implicit Mersenne exponent.) Many thanks.

Remarks and clarifications (see comments). Equivalently solve $y^2+Qz^2\equiv2\bmod{25964951}$ where $Q=674178680432399$, for odd integers $y,z\geq 1$. Here a Mersenne exponent is a integer $p$ (in fact can be proved that it is a prime number) such that $2^p-1$ is a prime number.

I hope that this version for a large Mersenne exponent is interesting for professors here, and we conclude that it is possible to do some work about the Question. If isn't interesting please add a comment, that I can to deleted the post.

Remarks. Wolfram Alpha calculator computed that $25964951^2-2$ is a prime number.

I would like to dedicate with all respect this post in the memory of persons killed in the Afghanistan earthquake at 22th of June.

References:

[1] Alexandru Gica and Florian Luca, On the Diophantine equation $2^x=x^2+y^2-2$, Funct. Approx. Comment. Math. 46(1): 109-116 (March 2012).

[2] Post edited on Mathematics Stack Exchange A diophantine equation inspired in a conjecture due to Gica and Luca (Jun 25, 2022).

Answer 1

The equation (1) for a fixed $x$ is equivalent to the congruence: $$y^2 \equiv 2(1+z^2)\pmod{x}.$$

For $x=25964951$, we have $2\equiv 3328351^2\pmod{x}$, and thus all solutions are obtained from those $z$ for which $1+z^2$ is a square modulo $x$ (there are $\frac{x-1}2$ such residue classes). Having such a $z$, we obtain all suitable values of $y$ as $y\equiv \pm 3328351\sqrt{1+z^2}\pmod{x}$.