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?

Wikipedia has this definition of prime constellation and only one type for $k=4$? — Dietrich Burde, Apr 05 '19 at 14:45
In the third example, we can insert $p+2$ which usually is ruled out in the definition of a prime constellation. The second example however is definitely a prime constellation of length $4$. — Peter, Apr 05 '19 at 19:45

score 3 · Accepted Answer · edited Jun 12 '20 at 10:38

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.

Are these all prime quadruplets?

1 Answers1