Elementary number-theoretical statements proved by Language Theory

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 $\mathbf{\text{ZFC}}$. 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.