Binary Additive Problems: Recursions for Numbers of Representations

TL;DR

This paper develops general recursive formulas to count the number of representations of positive integers as sums involving elements from increasing sequences, including classical problems like Goldbach and Lemoine-Levy conjectures.

Contribution

It introduces new recursive methods to compute the number of representations for various binary additive problems involving specific sequences.

Findings

Derived general recursions for binary representations

Applied recursions to classical problems like Goldbach and Chen

Provided formulas for counting representations in specific sequences

Abstract

We prove some general recursions for the numbers of representations of positive integers as a sum x+y, x in X, y in Y, where X,Y are increasing sequences. In particular, we obtain recursions for the number of the Goldbach, Lemoine-Levy, Chen and other binary partitions.