Studies in Additive Number Theory by Circles of Partition

In this paper we introduce and develop the circle embedding method. This method hinges essentially on a Combinatorial structure which we choose to call circles of partition. We provide applications in the context of problems relating to deciding on the feasibility of partitioning numbers into certain class 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 greater than $\frac{1}{2}$. This possibility could herald an unprecedented progress on class of problems of similar flavour. The paper finishes by giving a partial proof of the binary Goldbach conjecture.Comment: 43 pages; a partial proof of the binary Goldbach conjecture adde

Complex Circles of Partition and the Squeeze Principle

In this paper we continue the development of the circles of partition by introducing the notion of complex circles of partition. This is anenhancement of such structures from subsets of the natural numbers as base sets to the complex area as base and bearing set. The squeeze principle as a basic tool for studying the possibilities of partitioning of numbers is demonstrated.Comment: 12 page