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 Mersenne exponents $x\geq 13$ such that $x^2-2$ is a prime number.
Previous diophatine 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]. Mersene 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.
Question (Edited). I would like to know a Mersenne exponent $x>31$ with $x^2-2$ also a prime number, for which we are able to find or characterize all solutions of the corresponding equation $(1)$, over $k\geq 1$ integer, and over odd integers $y\geq 1$ and $z\geq 1$. Many thanks.
Example. I've calculated the first two solutions for $x=61$,
$$61k=y^2+3719z^2-2,$$ that are $(k;y,z)=(118;59,1)$ and $(126;63,1)$ I wondered for this $x$ or a different Mersenne exponent $x>31$ we can find all solutions (I'm interested in a case in which the set of solutions is not empty, as for $x=61$).
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).