The twin prime conjecture says there are infinitely many pairs $p,p+2$ that are both prime, and although we still don't know whether it's true there's been a lot of progress recently showing that there are infinitely many pairs $p,p+k$ that are both prime, for $k$ bounded by a constant.
It's also an old conjecture that there are infinitely many pairs $p,2p+1$ that are both prime (Sophie Germain primes and safe primes respectively). Is there any hope of adapting the recent twin prime progress to approximate the constant 2 in the Sophie Germain conjecture by something larger? That is, can it be proven that there are infinitely many pairs $p,kp+1$ that are both prime, with $k=O(1)$?
Of course Dirichlet and Linnik say that for every $p$ there is eventually a $k$ such that $pk+1$ is prime, but I'm more interested in the case of infinitely often rather than for all $p$, and I'd like $k$ to be significantly smaller than the proven bounds in Linnik's theorem. The best bound I know how to prove is $k=O(p)$, which follows from the existence of infinitely many "pseudo Sophie Germain primes", numbers $q$ that are prime or the product of two primes and for which $2q+1$ is prime (see arXiv:math/0603439). Letting $p$ be the largest prime factor of $q$ gives $k=2q/p\le 2p$. So I think any bound $k=o(p)$ would be interesting.