Conjecture: For all primes $p_n≥7$, there is at least one solution to the equation $$ p_n = p_k\# - p_m$$ where $p_k\#$ is any primorial, and $p_m$ is any prime number.
Has this been explored before? Can it been proven or disproven? I tried it numerically up to $ p_n =2322869$
Interestingly, the additive version of this, $$ p_n = p_k\# + p_m$$ seems to have solutions for most primes but not all; the lowest of which lacks a solution is $p_n=149$.
Partial answer : The above comments show that upto $14\cdot 10^6$ , there is a solution , the hardest case was the first prime that turned out to be tough. The primes are partially very big , so we will have to be content with probable primes.
What can we conclude now ? Hard to say. On the one hand , for every given prime $p>3$ , there are only finite many positive integers $n$ , for which $n$#$-p$ can be a prime number (we must have $n<p$ , otherwise $p$ is a nontrivial factor) and $n$# grows exponential.
On the other hand , the bigger the prime , the more chances we have to finally get a prime and that a prime factor of $n$#$-p$ with prime $p$ and $1<n<p$ must exceed $n$ increases the chance to get a prime , so it is hard to come to a final conclusion.
My guess is that there is a counterexample (a prime $p$ for which there is no solution) , but it might be very difficult to find it.