Are these all prime quadruplets?

Question

Wikipedia states on this page that $\{p, p+2, p+6, p+8\}$ is "the only prime constellation of length 4". But I believe this is not true, since $\{p, p+4, p+6, p+10\}$ and $\{p, p+6, p+12, p+18\}$ also satisfy the definition of a prime constellation (they do not form a complete residue class with respect to any prime). Moreover, Wolfram's Mathworld also states these other prime quadruplets as prime constellations on this page.

So is Wikipedia wrong?

Answer 1

You forgot the minimality requirement from the definition. For example the http://oeis.org/wiki/Prime_constellations makes the following distinction:

prime $k$-tuple - a strictly increasing sequence of $k$ primes ($p_1,p_2,\dots,p_k$) with $p_k-p_1=s(k)$, where $s(k)$ is not necessarily the smallest number $s$ for which there exist $k$ integers $b_1<b_2<\dots<b_k,b_k-b_1=s$, and for every prime $q$, not all the residues modulo $q$ are represented by $b_1,b_2,\dots,b_k$.

prime constellation, also called prime $k$-tuplet (not tuple) - a sequence of $k$ consecutive primes, i.e. $(p_1,p_2,\dots,p_k)$ with $p_k-p_1=s(k)$, where $s(k)$ is the smallest number $s$ for which there exist $k$ integers $b_1<b_2<\dots<b_k,b_k-b_1=s$ and, for every prime $q$, not all the residues modulo $q$ are represented by $b_1,b_2,\dots,b_k$.

Now both Wiki and MathWorld also mention the minimality in definitions of prime constellations. So if you accept these definitions, all of your three examples are prime $k$-tuples (quadruples), but only $(p, p+2, p+6, p+8)$ is a prime constellation (or prime quadruplet).

But I agree there is a confusion, for example because MathWorld at http://mathworld.wolfram.com/PrimeConstellation.html at some point says:

Another quadruplet constellation is $(p, p+6, p+12, p+18)$.

This I think is a mistake, because at its own definition of prime quadruplet (http://mathworld.wolfram.com/PrimeQuadruplet.html) which is referenced right at the beginning of the same paragraph, it mentions the minimality and that again the only example is $(p,p+2,p+6,p+8)$... Another confusion comes from wiki interchanging tuple and tuplet, and the minimality is only vaguely stated as "This represents the closest possible grouping", but if you go to https://en.wikipedia.org/wiki/Prime_k-tuple#Prime_constellations there you find slightly better "... an admissible prime $k$-tuple with the smallest possible diameter $d$ ...". Unfortunately tuple and tuplet seem to be quite frequently interchanged not only on wiki, so there is that...

There is also similar question here The definition of a prime constellation.

But in any way, the definitions are not dogma, different authors might use different definitions, it is probably best to check the definition used by particular article.