The definition of a prime constellation

Question

On Mathworld it is first stated that a prime constellation is a sequence of $k$ prime numbers, for which the gap between the last and the first minimizes. But later they show a table with prime constellations and cousin primes $(p,p+4)$ is said to be a prime constellation even though the gap is not minimized because of the 2-tuple $(p,p+2)$.

What is the exact definition of a prime constellation and is there some terminology for any sequence of primes whether the gap is minimized or not?

I appreciate any clarification because the definition seems to be confusing.

Answer 1

A tuple like $$[p,p+2,p+6]$$ is called a prime-constellation, if we cannot insert another number between the smallest and largest value without preventing that it can produce infinite many prime-tuples.

If we insert, for example $p+4$ in the above constellation, one of the numbers $p,p+2,p+4$ must be divisible by $3$, hence we cannot have infinite many prime-tuples of this form.

It is conjectured (but not proven even in the simplest cases) that prime-constellations produce infinite many prime-tuples.

Answer 2

A prime $k$-tuple is a repeatable pattern of primes that are as close as possible together. The prime $k$-tuple conjecture is also a special case of Dickson's conjecture, as it is the conjecture that for each positive integer $n$, there is an arithmetic progression of $n$ primes.

Prime constellations of length $k$ are then the shortest acceptable $k$-tuples of primes. That is, a $k$-tuple is valid unless there is a prime $q\le k$, which always divides the product of the terms.