For which $k$ with $0Because PARI/GP is not very fast in primilaty testing, I did not check the
pairs $k \times 3571\# \pm 1$ in ascending order, but I begun with $k=200,000$
and got the twin prime pair
$$210637\times 3571\# \pm 1$$


*

*Is there a $k$ with $0<k<210637$, such that $k\times 3571 \# \pm 1$ is a twin prime pair ?


I tried to sieve out small factors, but it turned out that many candidates
remain because no member of any pair can have a prime factor $\le 3517$.


*

*Can I speed up the search with PARI/GP by using some clever sieving methods ?


I don't know about PARI/GP, but NewPGen can sieve pretty fast.
Running the interval $0<k<200000$ for one minute eliminated about 170000 candidates.
https://primes.utm.edu/programs/NewPGen/
@Edit: I could have perhaps explained the programm a bit. It only sieves, for testing you need another program. Pfgw performs probable prime tests, and best thing is, it can use the files created by NewPGen.
http://sourceforge.net/projects/openpfgw/
