I'm looking for "long" arithmetic sequences that contain only prime (positive) numbers. For example: $$7,37,67,97,127,157$$
Is there any known bound for the length of these kind of sequences? Is it known that there exist arbitrarily long sequences?
I'm aware of Dirichlet's theorem. I also know that if $a_1$ is the first term and $d$ the difference of the sequence, then it should be $\gcd(a_1,d)=1$ and the length of the sequence is $\le a_1$ (because $a_1+a_1d$ is not prime).