# Prime chains with large gaps

It is well known that the gap between consecutive primes is unbounded. Is this still true for a chain of consecutive primes ?

More Formally : Is the following statement true for all natural numbers m and n ?

There are m consecutive primes $a_1,...,a_m$ , such that all the gaps are greater than n (this means $a_{k+1}-a_k>n$ for all k with 1 <= k <= m-1) ?

I also heard about primes in arithmetic progressions, but I always wondered if the primes must be consecutive in such progressions.

Can any of the known properties of the prime-numbers help to answer this question ?

Yes, this is known. It feels a bit like squashing a fly with a B-52, but there is a beautiful theorem of Shiu (Strings of Congruent Primes, 2000) which seems to cover everything you want. A direct consequence is that for any $n,m$ the following is true:
There are $m$ consecutive primes $a_1,\ldots,a_m$ that are all congruent to $1$ mod $n$.
So not only are the gaps between these primes at least $n$, but they all lie within a prescribed arithmetic progression. (It's important not to confuse this with $a_1,\ldots,a_m$ forming an arithmetic progression: a result of such strength is not yet known.)