Why does the primorial $23\#$ come up so often in long prime arithmetic progressions? This section of the Wikipedia article on the Green-Tao theorem gives examples of the longest known arithmetic progressions of prime numbers. For every known arithmetic progression of $24$ or more consecutive primes, the common difference is a multiple of the primorial $23\# = 223,092,870$ (the product of all primes up to $23$).
Is it a coincidence that $23$ is the largest prime less than the length of the long sequences? With enough computing power, would we eventually expect to find arithmetic sequences of $29$ or more primes whose common differences are multiples of $29\#$? If not, is there something special about the number $23\#$, or does it simply reflect the limits of our computing power?

Given an arithmetic progression $a,a+d,\ldots, a+(n-1)d$ and a prime $p\le n$ with $p\nmid d$, at least one of the $a+kd$ will be a multiple of $p$.
