# Consecutive prime terms in an arithmetic sequence

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?

Background

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).

• Yes , this is known as the Green-Tao theorem : en.wikipedia.org/wiki/Green%E2%80%93Tao_theorem – user252450 Jan 20 '16 at 16:50
• If the primes must be consecutive, I do not know whether it is only conjectured or proven. If the primes are arbitary, you can construct arbitary long sequences. – Peter Jan 20 '16 at 17:01
• @Peter The example included is not made of consecutive primes ;) – N. S. Jan 20 '16 at 17:05
• @Peter Is the variant about consecutive primes really conjectured to be true ? I didn't heard about it but it seems very unlikely that for some $n$ we will have : $$g_n=g_{n+1}=\ldots=g_{n+10^{100}}$$ where $g_n=p_{n+1}-p_n$ is the prime gap . But , who knows...maybe it's true (that would be mind-blowing ) – user252450 Jan 20 '16 at 17:13
• The gap should be a primorial. It does not work for an arbitary gap. – Peter Jan 20 '16 at 17:17