Denote $p$#$:=2\cdot 3\cdot 5\cdot 7\cdots p$
What is the smallest twin-prime of the form $k\cdot 11699$#$\pm 1$ , where $k$ is a positive integer ?
Sieving out the candidates with Newpgen and checking with PFGW , I found no $k\le 10^5$ doing the job. How large can the smallest $k$ be expected ?