What is the set of positive integers $n$ such that $(6k+2)^n + 3$ and $(6k+2)^n + 5$ are both prime? That is, for some fixed $k\in N$, what is the set of positive integers $n$ such that $(6k+2)^n+3$ and $(6k+2)^n+5$ are both prime? At least, is this set finite?
See if you want the entire thing prime then the term should not have any prime factors. So $(6k+2)^n$ should not be divisible by 3 or 5. This term can never be divisible by 3, but can be divisible by 5. So k should not be 3, 8, 13, and so on.

I cannot think how can i find the set for n, but i think this might help.
See if you want the entire thing prime then the term should not have any prime factors. So $(6k+2)^n$ should not be divisible by 3 or 5. This term can never be divisible by 3, but can be divisible by 5. So k should not be 3, 8, 13, and so on.

I cannot think how can i find the set for n, but i think this might help.