Please help me proved or disprove the conjecture below. Thanks.
For every (fixed) prime $p$, there are infinitely many integers $k$ such that $p$, $p+k$, and $kp+1$ are all prime?
I wasn't exactly sure if this was implied by Dickinson's Conjecture, since two variables are required. It seems likely that it is the case, and is proven true that there are infinitely many primes for $p$ and $p+k$ for a fixed prime $p$ (Euclid's Theorem) as well as $kp+1$ (Dirichlet's Theorem), for any prime $p$. Would a combination of both theorems prove the conjecture above?