Semigrupos numéricos e equações diofantinas: uma relação entre elementos gerados e quantidade de soluções

Abstract

This work presents an introduction to Number Theory from the perspective of Diophantine Equations and, subsequently, to the study of Numerical Semigroups associated with these equations. Initially, we develop fundamental results on Diophantine equations and numerical semigroups, allowing the characterization of some Numerical Semigroups, in particular, those with maximal embedding dimension, with the Arf property, saturated, and irreducible. These results may be relevant in a future attempt to solve the problem of finding the Frobenius number for semigroups generated by three elements and, in particular, to formalize a counting principle for the number of solutions of equations with three variables. Furthermore, an original result will be developed that counts the number of solutions of an equation with two variables, which will be compared with other known results, and its extension to three variables is suggested, optimizing existing algorithms in particular cases.