Longest Sequence of Consecutive Primes in Arithmetic Sequence

Sequence
The longest known sequence of consecutive prime numbers in arithmetic progression starts at $121 \, 174 \, 811$, has a length of $6$ and a common difference of $30$:


 * $121 \, 174 \, 811, 121 \, 174 \, 841, 121 \, 174 \, 871, 121 \, 174 \, 901, 121 \, 174 \, 931, 121 \, 174 \, 961$

Proof
But note that $121 \, 174 \, 811 + 6 \times 30 = 121 \, 174 \, 991 = 7^2 \times 2 \, 472 \, 959$ and so is not prime.

Inspection of tables of primes (or a computer search) will reveal that no longer such sequences are known.