Does this conjecture have a name ?
I assume it is true but has not been proven or disproven.
I am aware of prime twins, Dickson's conjecture, Bunyakovsky conjecture, Schinzel's hypothesis H, Bateman-Horn conjecture etc
See :
https://en.wikipedia.org/wiki/Dickson%27s_conjecture
https://en.wikipedia.org/wiki/Bunyakovsky_conjecture
https://en.wikipedia.org/wiki/Schinzel%27s_hypothesis_H
https://en.wikipedia.org/wiki/Bateman%E2%80%93Horn_conjecture
But this is a bit different ;
"mick's prime power conjecture"
Let $p$ be a prime.
Then there is always an integer $n$ such that
$q_n = p^n - 2$
and $q_n$ is also a prime.
Not important but for those who are curious, My mentor believes the weaker conjecture what inspired me :
"tommy's prime power conjecture "
Let $p$ be a prime such that $p-2$ is also a prime.
Then there is always an integer $n>1$ such that
$q_n = p^n - 2$
and $q_n$ is also a prime.
This probably has no name.
Related is this sequence : primes $p$ such that $p^2 - 2$ is also prime. So essentially the case $n = 2$.
and it starts like $2,3,5,7,13,19,29,37,43,47,61,...$
(this seems a bit biased towards the largest of prime twins for small primes, for those interested. It probably goes away for larger values. But it might relate to tommy's conjecture somewhat)
I note that there seem to be infinitely many primes of the form $2^m - 1$ (Mersenne) or $3^m - 2$.
Back to my conjecture :
Here are some examples listed by increasing primes and minimum value $n$ giving a prime :
$$2^2 - 2 = 2$$
$$3^2 - 2 = 7$$
$$5^1 - 2 = 3$$
$$7^1 - 2 = 5$$
$$11^4 - 2 = 14639$$
$$13^1 - 2 = 11$$
$$17^6 - 2 = 24137567$$
$$19 - 2 = 17$$
$$23^{24} - 2 = 480250763996501976790165756943039$$
$$29^2 - 2 = 839$$
$$31^1 - 2 = 29$$
$$37^2 - 2 = 1367$$
$$41^4 - 2 = 2825759$$ ...
In fact we dare to make our conjectures stronger ; there are infinitely many $n$ for each prime $p$.
I (or we) are probably not the first to conjecture or observe this. Hence I wonder if it has been named yet. Any papers on it would be nice too ofcourse.
edit
A strange phenomenon but maybe just coincidence :
Due to the list above it seems we get all prime powers being of the form
$p^1 = p$ or $p^{2k}$
Why so many even powers ?
So I decided to extend the list :
$$43 - 2 = 41$$
$$ 47^2 - 2 = 2207$$
$$ 53^4 - 2 = 7890479$$
$$59^4 - 2 = 12117359$$
$$61^2 - 2 = 3719$$
$$67^3 - 2 = 300761$$
so finally the trend is broken at $67$.
An explaination I have not.
edit 2
I found these related sequences :
Updated: This sequence and its comment section answer the present question: OEIS sequence A267944. Thus, both the stated conjectures are very likely to be true (again, see the comments of $A267944$).
Now, an interesting additional sequence would be constructed by sorting in ascending numerical order the smallest values of ${p_n}^k-2$ for $n=1,2,3,\ldots$ ($k:=k(p_n) \in \mathbb{Z}^+$). Since every term of $a(n)$ such that $n>1$ has to be even, we get only the following candidates and we can easily check them one by one in a few minutes.
Here are the only sequences of the OEIS containing the subsequence $2,3,5,7,11,17,29$ and whose next term is an odd number: $A238528$, $A113192$, $A062294$, $A127272$, $A196375$, $A356627$.
Then, I verified that $a(8)=41$ and we know that $43$ cannot be a term since $45$ is not a prime. Moreover, the prime number $71$ is a term since $73$ is a prime, and consequently the only OEIS sequence that cannot easily be disregarded is $A238528(n):= 2, 3, 5, 7, 11, 17, 29, 41, 59, 71, 101, 107, 137, 149,\ldots$