Studies in Additive Number Theory by Circles of Partition

TL;DR

This paper introduces the circle embedding method using circles of partition to analyze additive number theory problems, enabling partitioning of large numbers into sets with density over 50%, and applies it to Goldbach and Lemoine conjectures.

Contribution

The paper develops a novel geometric combinatorial method called circles of partition, advancing the analysis of number partitioning problems and conjectures.

Findings

Able to partition sufficiently large numbers into sets with density > 1/2

Provides an asymptotic proof of Goldbach and Lemoine conjectures

Introduces the circle embedding method as a new tool in additive number theory

Abstract

In this paper, we introduce and develop the circle embedding method. This method hinges essentially on a combinatorial-geometric structure which we choose to call circles of partition. We provide applications in the context of problems that relates to deciding on the feasibility of partitioning numbers into certain subset of integers. In particular, our method allows us to partition any sufficiently large number $n\in\mathbb{N}$ into any set $\mathbb{H}$ with natural density strictly greater than $\frac{1}{2}$. This possibility could herald an unprecedented progress on categories of problems of similar flavour. The paper finishes by presenting an asymptotic proof of the binary Goldbach and Lemoine conjecture as an application of the developed method.