A Complete Congruence System for the Erdos-Straus Conjecture

Abstract

In this paper we attack the Erdos-Straus conjecture by means of the structure of its solutions, extending and improving the results of a previous paper. Using previous results and supported by the works of Elsholtz and Tao and Monks and Velingker we define a system of congruences for which there are always solutions to the Erdos-Straus conjecture and which we conjecture to include all prime numbers. For this purpose, and always taking into account a result due to Mordell that limits the congruences admitting polynomial identities to those that are not quadratic residues, we will adopt a transversal approach and classify the solutions by their form and not by those congruences that produce them. Thus we define two new types of solutions, which we call Type A and B, and relate them to the already known Type II solutions and study their properties. Finally we conjecture that every prime number has at least one solution of Type A or B and we associate a congruence and a general polynomial to each Type of solution.