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](https://en.wikipedia.org/wiki/Sophie_Germain_prime) 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](https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions) and [Linnik](https://en.wikipedia.org/wiki/Linnik%27s_theorem) 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](http://arxiv.org/abs/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.



Primes $p$ for which $pk+1$ is prime for small $k$ (or approximating Sophie Germain)
[This paper](http://www.ams.org/journals/mcom/2005-74-252/S0025-5718-05-01749-7/S0025-5718-05-01749-7.pdf#page=2) uses "$\hspace{.01 in}P(n)$ to denote the greatest prime factor of an integer $n\hspace{.01 in}$",
  

and obtains the following result as Theorem 2 on page 2.



> 
> Let $a\in \mathbb{Z},\theta < 1-\frac12 \exp \left(-\frac14 \right) = 0.6105\cdots$ .
>    
> 
>  Then there exist effectively computable constants $X\_1(a,\theta),\delta(\theta)>0$, such that, if $x > X\_1$, we have $$\displaystyle\sum\_{\substack{p\leq x \\ P(p+a)>x^{\theta}}} 1 > \delta(\theta) \frac{x}{\log x}.$$
> 
> 
> 


  

In particular, this implies that for all $b$, $\:\:$ if $\;\; \frac{\exp\left(-\frac14 \right)}{2-\exp\left(-\frac14 \right)} \: < \: b \;\;$ then there are
  

infinitely many pairs $\;\; p\:,\:p\hspace{-0.04 in}\cdot\hspace{-0.04 in}k+1 \;\;$ that are both prime, with $\:\: k < p^b \;\;$.


Since $\; \frac{\exp\left(\hspace{-0.02 in}-\frac14 \hspace{-0.03 in}\right)}{2\hspace{.02 in}-\hspace{.03 in}\exp\left(\hspace{-0.02 in}-\frac14 \hspace{-0.03 in}\right)} < \frac23 < 1 \;$, $\;$ that gives a bound $\: k\in o(\hspace{.03 in}p) \;$.


  
(Incidentally, I only knew about that paper due to Mark's answer to my question [here](https://mathoverflow.net/q/130429).)
  
  



