# Finite number of consecutive prime numbers

Suppose we have $$n$$ consecutive prime numbers $$p_k$$ with $$n \in\mathbb{N}$$ such that $$p_k-p_{k-1}=m$$ and $$k=2..n$$. Is it possible to find $$m$$ and $$n$$ in order to have a finite number of these consecutive primes? For example, if we put $$m=6,n=3$$, for the first $$23$$ primes, we get only the triplet $$(47,53,59)$$ satisfying this condition. Obviously, for example, for $$n=3,m=2$$ (twin primes), there are only three consecutive primes $$(3,5,7)$$. Is this the only case we can have a finite number of consecutive primes with $$n\gt 2$$? Thanks.

• You mean $m=2$ for the twin primes? Beyond that, I'm not sure what you are asking. Pick some prime $p$ not dividing $m$ and work $\pmod p$. That gives you a bound for the chain.
– lulu
Commented Nov 17, 2022 at 12:30
• There is another trio below 100. Then there is 251,257,263,269 Commented Nov 17, 2022 at 12:40
• Are you sure you want the n primes to be consecutive? Commented Nov 17, 2022 at 12:42
• The Green Tao theorem guarantees arbitary long arithmetic progressions consisting of only prime numbers, but this theorem is not proven for consecutive primes. Commented Nov 17, 2022 at 13:18
• Please edit your post for clarity. As it stands, I can't sort out what you are asking.
– lulu
Commented Nov 17, 2022 at 13:51

For consecutive primes larger than $$5$$ in an arithmetic progression, the common increment must be a multiple of $$6$$, as those primes have a form $$6t\pm 1$$. Any five consecutive positive integers of the form $$6t-1$$ or $$6t+1$$ (or $$6st \pm 1$$ where $$5 \not \mid s$$) will have one member that is divisible by $$5$$, limiting sequences of primes to at most $$4$$ in length.
You can address this by making $$5$$ a factor of the increment, looking for strings among integers of the form $$30t \pm r$$ where $$r \in \{1,7,11,13\}$$, but then strings of $$7$$ such numbers will contain at least one member divisible by $$7$$. You could move to numbers of the form $$210t \pm r$$, but you will appreciate that as the increment of the arithmetic sequence gets larger, the likelihood of finding strings of consecutive primes diminishes.