Exceptional autonomous components of Goldbach factorization graphs

TL;DR

This paper introduces Goldbach factorization graphs (GFGs) to analyze the binary Goldbach conjecture, identifying special autonomous components that relate to potential counterexamples and exploring their properties computationally.

Contribution

The paper defines GFGs, proves the existence of unique exceptional autonomous components, and provides computational evidence of their properties up to 10^8, linking graph theory to prime number conjectures.

Findings

Existence of exactly one EAC induced by two vertices.

Six EACs found for n ≤ 10^8 at specific values.

Repository of GFG drawings and properties created.

Abstract

We introduce a concept of a Goldbach factorization graph (GFG) $F_n$, which can be constructed for each even integer $n$ greater than 2. We prove that, if $n$ does not satisfy the binary Goldbach conjecture (BGC), then $F_n$ contains a special source strongly connected component (exceptional autonomous component, EAC). We analyse existence and properties of EACs using deductive and computational approaches. In particular, we prove that there exists exactly one EAC induced by two vertices. Using computer-aided search, we show that for $n \leq 10^8$ there are 6 EACs, each inside a different GFG, and they are located at the relative beginning of the checked range, namely, for $n\in\{128,1718,1862,1928,2200,6142\}$. Using classic graph algorithms, the constraint programming method, and metaheuristic approaches, we have prepared a repository of drawings and some selected properties of the found EACs and GFGs which contain them. The concept of EAC relates to the BGC, but more generally, it represents interesting self-conjugation of prime numbers under a relation which combines addition and multiplication.