A primitive root $h$ of $n$ is a generator of the cyclic modulo multiplicative group $\mathbb{Z}^\times_n$.

Suppose, $\mathbb{P}_{\langle 2\rangle}=\{p_i \mid \langle 2\rangle=\mathbb{Z}^\times_{p_i}\}$ denotes the set of all primes that have a primtive root $2$. Let, $2^k\equiv x_i \mod{p_i}$, for some fixed even integer $k\gg\log{p_i}$ for all $p_i \in\mathbb{P}_{\langle 2\rangle}$.

My questions are,

1. Is there any upper and lower bound on each $x_i$, even asymptotically? or do they have some special form?
2. Is there any relation between any arbitrary pair $(x_i,x_j)$ for $i\ne j$?

In short, is it possible to predict $x_i$ even before $p_i$ is specified?

A primitive root $h$ of $n$ is a generator of the cyclic modulo multiplicative group $\mathbb{Z}^\times_n$.

Suppose, $\mathbb{P}_{\langle 2\rangle,N}=\{p_i <N\mid \langle 2\rangle=\mathbb{Z}^\times_{p_i}\}$ denotes the set of all primes $p_i<N$ that have a primtive root $2$. Let, $2^k\equiv x_i \mod{p_i}$, for some fixed even integer $k\gg\log{p_i}$ for all $p_i \in\mathbb{P}_{\langle 2\rangle,N}$.

My questions are,

1. Is there any upper and lower bound on each $x_i$, even asymptotically? or do they have some special form?
2. Is there any relation between any arbitrary pair $(x_i,x_j)$ for $i\ne j$?

In short, is it possible to predict $x_i$ even before $p_i$ is specified?