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 $73 - 6^2 = 6^3 - 179$
For every odd prime $\ p,\ $ and for every integer $\ n>1\ $ there exists exactly one positive integer $\ m=\mu(p\ n)\ $ such that $$ n^m\ <\ p\ \le\ n^{m+1}. $$
Odd prime $\ p\ $ is said to be $\ n$-doubtful (or $n$-questionable) $\,\ \Leftarrow:\Rightarrow\,\ q:=n^m+n^{m+1}-p\,\ $ is not a prime, where $\ m=\mu(p\ n).$
Odd prime $\ p\ $ is trustful $\,\ \Leftarrow:\Rightarrow\,\ $ there exists integer $\ n>1\ $ such that $\ p\ $ is not $n$-doubtful. Otherwise, $p$ mistrusted.
The OP's conjecture (actually a question) was that every odd prime is trusted.
The first $2$-questionable prime is $\ p=23\ $ because
$$ 2^4 < p\ < 2^5 $$
and
$$ q\ :=\ 2^4+2^5\ -\ p\ =\ 25 = 5^2 $$
is not a prime. Thus, there remains to verify primality (or non-primality) of
$$ n^m + n^{m+1} - 23 $$
just for a very limited number of cases, for all $\ n>2\ $ such that $\ n<23$:
The next $2$-questionable prime is $\ 41.\ $ However,
$$ 67\ =\ 3^3+3^4-41 $$
is a prime. The conjecture holds for $\ p:=41.$
There is just one more $2$-questionable prime in the $\ [2^5;2^6]\ $ range, namely $\ p:=47.\ $ But $\ 47\ $ is not $3$-doubtful -- indeed:
$$ 61\ = 3^3+3^4-47\ $$
is a prime. Next, we get a $2$-doubtful prime $\ 127\ $ from the higher end of of $\ [2^6;2^7].\ $ It's not $3$-doubtful though:
$$ 197\ =\ 3^4+3^5-127 $$
is a prime.
Given a huge list of consecutive primes, a computer program can verify the conjecture quickly within the given range of primes.
I am still curious about the smallest prime that is both $2$- and $3$-doubtful, as well as about
the general question about prime $\ d_n\ $ which is the smallest among $k$-doubtful for every $\ k\le n.$
REMARK doubtful = questionable (but of course :) ).
Actually:
The smallest $2$- and $3$-questionable prime is $\ p:=73.$
Indeed, $$ 119 = 2^6+2^7-73 $$ is not a prime ($119=7\cdot 17$) hence prime $73$ is $2$-questionable. Also, $$ 35 = 3^3+3^4-73 $$ is not a prime hence prime $73$ is $3$-questionable. Great!*
Furthermore, the same prime $\ p:=73\ $ is also $4$-questionable since
$$ 247 = 4^3+4^4-73 $$
is not a prime $\ (247=13\cdot 19),\ $ as well as $5$-questionable:
$$ 77 = 5^2+5^3-73 $$
is not prime. Thus,
Prime $\,73\ $ is the smallest that is $2$- and $3$- and $4$-questionable. Furthermore, prime $\,73\ $ is the smallest that is $2$- and $3$- and $4$- and $5$-questionable.
However, the conjecture holds for prime $73$ since it is not $6$-questionable; indeed:
$$ 179 = 6^2+6^3-73 $$ is a prime.
