I want to find a prime $p$ such that $pn \pm 1$ is prime for some fixed $n$.
Examples of $n$ are $1683$ or $99617$. Any $p$ will do, it doesn't need to be the smallest possible value.
I have tried bruteforcing a few examples. My method is to try successively larger values of $p$ until a prime is produced. However, for some values of $n$ I have tried primes up to $100,000,000$ (one hundred million) and still not found an answer.
I've tried searching online, but I haven't found an answer. I'm not sure I have the right terminology. The closest result I've found are Sophie Germain primes, which are the special case of $n=2$.
EDIT: To be clear, my question is "Is there a general method that I can plug a value of $n$ into and it will produce a prime $p$ so that the above is satisfied?"