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 ?


It's not NewPGen with nice sieving of these forms, but using the simple is_prime($n-1) && is_prime($n+1) method took Perl/ntheory 20 minutes on my laptop to find k=45593, 9 minutes more for k=65996, 47 seconds more for k=67724, ...
In all, 1.5 hours for the whole range to get:
45593, 65996, 67724, 120451, 127964, 210637

The NewPGen executables don't work on my Macbook, and 282 doesn't compile.  Anyone know of something newer than 282 or fixes for 64-bit (it's trying to store a pointer in a int)?
Ideas for sieving $k * n\# \pm a$ would be nice to see.
