What is a prime constellation?



What is a prime constellation?

In: Mathematics

It’s not easy, but I’ll try.

A prime constellation is also known as a prime k-tuplet. It’s a list of k consecutive prime numbers that verify some property of being “as close as possible” in some sense.


To define this sense of “closeness”, first I need to define s(k). s(k) is the smallest number s such that there exists k natural numbers b_1,b_2,…,b_k verifying :

* b_1 < b_2 < … < b_k
* b_k – b_1 = s
* For each prime number q, not all residues modulo q are represented by b_1,…,b_k.

The last property is the most complicated, obviously. A “residue modulo q” is basically the remainder of the division of b_i by q. Since it’s a remainder, all the possible residues modulo q are 0,1,2,…,q-1. This means that b_1,…,b_k are “invalid” if for some prime q, the set of the remainders of b_1,…,b_k divided by q contains every number from 0 to q-1 (not necessarily in order).

In particular (assuming k large enough) it means that if b_1 is even and thus its residue modulo 2 is 0, then all other b_i must be even too to avoid having one of them with a residue 1 (which is the only possible other residue modulo 2).

You can also note that any prime q>k cannot have all its residues represented by b_1,…,b_k anyway, so this property is only relevant for few prime numbers.


Now that s(k) is defined, we can define a prime k-tuplet.

A prime k-tuplet is a sequence of k consecutive prime numbers p_1,…,p_k such that :

* p_k – p_1 = s(k)
* For each prime number q, not all residues modulo q are represented by p_1,…,p_k.

In other words, it’s a set of primes that verifies this “not all residues” property while being the “closest possible” since the different between the last p_k and first p_1 is as small as any sequence of number (not necessarily prime !) verifying this property can ever be.


I hope this helps