S-invariant and S-multinvariant functions and some symmetry groups of algebraic sieves

TL;DR

This paper introduces algebraic sieves, invariant functions, and symmetry groups to analyze sieves like the Goldbach sieve, revealing new symmetries and group actions related to prime number sets and conjectures.

Contribution

It defines algebraic sieves and invariant functions, and studies their symmetry groups, applying these concepts to the Goldbach sieve with dihedral groups.

Findings

Identified symmetry groups of invariant functions in algebraic sieves.

Connected symmetry groups to subgroups of the ring of integers modulo N.

Applied the framework to the Goldbach sieve, revealing algebraic structures.

Abstract

In this article we introduced algebraic sieves, i.e. selection procedures on a given finite set to extract a particular subset. Such procedures are performed by finite groups acting on the set. They are called sieves because there are certain sets of numbers which, with appropriate groups, can select, for example, a set of primes, think of the famous Eratosthenes sieve. In this article we have given a general definition of algebraic sieves. And we also introduced the notion of invariant and multi-invariant functions, certain permutations on the sieve set which, in the invariant case, commute with the action of a given sieve-selecting group and the automorphism of that group, and multi-invariants which commute with all groups and their respective automorphisms. By means of such functions we have given symmetries on such sieves. In particular, we studied certain groups of symmetries of invariant functions. Then, using such notions, we studied a particular example, the Goldbach sieve, where the selector groups are dihedral groups and the selected set consists of the primes and, in some cases, the numbers $1$ and $N-1=p$, with $N$ being even and p prime, which satisfies the Goldbach conjecture for $N$. We have shown that one of these symmetry groups is isomorphic to a subgroup or affinity group of the ring of integers modulo N with N an integer even $\mathbb{Z}_{N}$.