Elementary number-theoretical statements proved by Language Theory

TL;DR

This paper presents a novel approach to prove elementary number-theoretical statements using formal language relationships, bridging Language Theory with traditional proof methods in set theory.

Contribution

It introduces a framework to derive number-theoretical theorems via formal languages and shows how these proofs can be converted into standard ZFC proofs.

Findings

Proved theorems about densely divisible numbers

Derived results on semi-perimeters of Pythagorean triangles

Established proofs for partitions into consecutive parts

Abstract

We introduce a method to derive theorems from Elementary Number Theory by means of relationships among formal languages. Using $\sigma$-algebras, we define what a proof of a number-theoretical statement by Language Theory means. We prove that such a proof can be transformed into a traditional proof in $\textbf{ZFC}$. Finally, we show some examples of non-trivial number-theoretical theorems that can be proved by formal languages in a natural way. These number-theoretical results concern densely divisible numbers, semi-perimeters of Pythagorean triangles, middle divisors and partitions into consecutive parts.