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

Francesco Maltese

TL;DR

This work develops an algebraic, group-action perspective on numerical sieves by introducing S-invariant and S-mult invariant functions to study sieve symmetries. It builds a formal framework for algebraic sieves, including dihedral and Goldbach variants, and derives structural descriptions of the symmetry groups $\widehat{Aut(G)}^{S}$, with explicit results for dihedral actions on $\mathbb{Z}_{2n}$ in both odd and even cases. The paper provides detailed calculations and criteria to determine the calculation group $\mathcal{G}_N$ and applies these to concrete examples, culminating in conjectures that connect sieve symmetries to Goldbach-type phenomena. Overall, it links finite-group symmetries, automorphism structures, and orbit data to the design and analysis of algebraic sieves, offering a novel algebraic lens on classic prime-related conjectures and their reformulations. The results may inform both theoretical understanding and algorithmic approaches to sieve-based selections via symmetry considerations.

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 $\mathbb{Z}_{N}$.

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

TL;DR

This work develops an algebraic, group-action perspective on numerical sieves by introducing S-invariant and S-mult invariant functions to study sieve symmetries. It builds a formal framework for algebraic sieves, including dihedral and Goldbach variants, and derives structural descriptions of the symmetry groups , with explicit results for dihedral actions on in both odd and even cases. The paper provides detailed calculations and criteria to determine the calculation group and applies these to concrete examples, culminating in conjectures that connect sieve symmetries to Goldbach-type phenomena. Overall, it links finite-group symmetries, automorphism structures, and orbit data to the design and analysis of algebraic sieves, offering a novel algebraic lens on classic prime-related conjectures and their reformulations. The results may inform both theoretical understanding and algorithmic approaches to sieve-based selections via symmetry considerations.

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 and , with being even and p prime, which satisfies the Goldbach conjecture for . 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 .

Paper Structure

This paper contains 12 sections, 62 theorems, 258 equations.

Table of Contents

  1. Introduction
  2. S-invariant functions
  3. Permutations induced
  4. The algebraic structure of $\widehat{{Aut(G)}^{S}}$
  5. Examples of $\widehat{{Aut(G)}^{S}}$
  6. Algebraic sieves
  7. Examples of algebraic sieves
  8. Multi-invariant functions
  9. Some symmetry groups of an algebraic sieve
  10. The group $\widehat{Aut(D_n)}^{\mathcal{C}(D_{p,q}^{2n}; x_{0,q} )}$
  11. Methods of calculation for the group $\mathcal{G}_{N}$ .
  12. Final conclusions and conjectures

Key Result

Proposition 1.3

If f $\in Sym(X)$ (i.e set of bijective applications from X to itself) is a $\phi-invariant$ then $\phi(Stab_{G}(x_{0}^{i}))=Stab_{G}(x_{0}^{j})^{g_{ij}}$ fixed $x_{0}^{i}$ for $i=1,2,\dots,t$

Theorems & Definitions (119)

  • Definition 1.1
  • Definition 1.2
  • Proposition 1.3
  • Proposition 1.4
  • proof
  • Proposition 1.5
  • proof
  • Proposition 1.6
  • proof
  • Corollary 1.7
  • ...and 109 more