In 2004 Ben Green and Terence Tao proved that for any positive integer k, there exist arithmetic progressions of primes with k terms in the following historical paper:
Ben Green and Terence Tao, The primes contain arbitrarily long arithmetic progressions, Annales of Mathematics, 167 (2008), pp.481-547, doi: 10.4007%2Fannals.2008.167.481.
We find arithmetic progressions of length 6-11 by using MATLAB.