Additive Number Theory via Approximation by Regular Languages

TL;DR

This paper introduces a novel approach combining automata theory with additive number theory, proving new theorems such as representing numbers as sums of binary strings with equal zeros and ones.

Contribution

It presents a new method that applies formal language techniques to solve problems in additive number theory, demonstrating its effectiveness with specific theorems.

Findings

Every number greater than 25 can be expressed as the sum of at most three numbers with balanced binary representations.

New theorems in additive number theory are established using automata and formal languages.

The approach bridges automata theory and number theory, opening new research avenues.

Abstract

We prove some new theorems in additive number theory, using novel techniques from automata theory and formal languages. As an example of our method, we prove that every natural number > 25 is the sum of at most three natural numbers whose base-2 representation has an equal number of 0's and 1's.