Is it true that every prime number is a member of one or more pairs of primes $(p_1, p_2)$ such that $p_1$ and $p_2$ are inbetween and equidistant from two adjacent powers of some number $n$? i.e.

$0 < p_1 - n^m = n^{m+1} - p_2 $ for some positive integers $n, m$

For example the prime number $73$ is a member of the pair $(73, 179)$ where $73 - 6^2 = 6^3 - 179$

Is it true that every odd prime number is a member of one or more pairs of primes $(p_1, p_2)$ such that $p_1$ and $p_2$ are inbetween and equidistant from two adjacent powers of some number $n$? i.e.

$0 < p_1 - n^m = n^{m+1} - p_2 $ for some positive integers $n, m$

For example the prime number $73$ is a member of the pair $(73, 179)$ where $73 - 6^2 = 6^3 - 179$

Is it true that every prime number is a member of one or more pairs of primes $(p_1, p_2)$ such that $p_1$ and $p_2$ are equidistant from and between two adjacent powers of some number $n$? i.e.

$p_1 - n^m = n^{m+1} - p_2 = d$ for some positive integers $n, m, d$

For example the prime number $73$ is a member of the pair $(73, 179)$ where $73 - 6^2 = 6^3 - 179 = 37$

Is it true that every prime number is a member of one or more pairs of primes $(p_1, p_2)$ such that $p_1$ and $p_2$ are inbetween and equidistant from two adjacent powers of some number $n$? i.e.

$0 < p_1 - n^m = n^{m+1} - p_2 $ for some positive integers $n, m$

For example the prime number $73$ is a member of the pair $(73, 179)$ where $73 - 6^2 = 6^3 - 179$

Is it true that every prime number is a member of one or more pairs of primes $(p_1, p_2)$ such that $p_1$ and $p_2$ are equidistant and between two adjacent powers of some number $n$? i.e.

$p_1 - n^m = n^{m+1} - p_2 = d$ for some positive integer $d$

For example the prime number $73$ is a member of the pair $(73, 179)$ where $73 - 6^2 = 6^3 - 179 = 37$

Is it true that every prime number is a member of one or more pairs of primes $(p_1, p_2)$ such that $p_1$ and $p_2$ are equidistant from and between two adjacent powers of some number $n$? i.e.

$p_1 - n^m = n^{m+1} - p_2 = d$ for some positive integers $n, m, d$

For example the prime number $73$ is a member of the pair $(73, 179)$ where $73 - 6^2 = 6^3 - 179 = 37$