In this post we denote the Euler's totient function as $\varphi(n)$, first we show a claim related to Mersenne primes, see for example this Wikipedia and secondly we are going to ask a related conjecture.
We denote as $m=2^p-1$ a Mersenne prime, and we assume that there exists $k\geq 1$ an integer satisfying that $M=2^{p+2k}-1$, whenever $m<M$ are distinct Mersenne primes. With these definitions it is possible to prove the following claim.
Claim. If $m$ is a Mersenne prime and $1\leq k$ is an integer such that $2^{2k}m+2^{2k}-1$ is also a Mersenne prime, then the pair $(m,k)$ is a solution of the equation
$$(m^2-1)\left((2^{2k}m+2^{2k}-1)^2-1\right)=16\varphi\left(m(2^{2k}m+2^{2k}-1)\right)\varphi(\frac{m+1}{2})\varphi\left((m+1)2^{2k-1}\right).$$
Question. Prove or refute, showing a counterexample, the following conjecture:
Let $1\leq z$ an integer, and $1\leq\kappa$ also integer satisfying $$(z^2-1)\left((2^{2\kappa}z+2^{2\kappa}-1)^2-1\right)=16\varphi\left(z(2^{2\kappa}z+2^{2\kappa}-1)\right)\varphi(\frac{z+1}{2})\varphi\left((z+1)2^{2\kappa-1}\right).$$ Then $z$ is a Mersenne prime, and $2^{2\kappa}z+2^{2\kappa}-1$ is also a Mersenne prime.
Many thanks.