Some Results on Integer Solutions of Quadratic Polynomials in Two Variables

Abstract

Although it is true that there are several articles that study quadratic equations in two variables, they do so in a general way. We focus on the study of natural numbers ending in one, because the other cases can be studied in a similar way. We have given the subject a different approach, that is why our bibliographic citations are few. In this work, using basic tools of functional analysis, we achieve some results in the study of integer solutions of quadratic polynomials in two variables that represent a given natural number. To determine if a natural number ending in one is prime, we must solve equations (i) <img src=image/13422919_01.gif>, (ii) <img src=image/13422919_02.gif>, (iii) <img src=image/13422919_03.gif>. If these equations do not have an integer solution, then the number P is prime. The advantage of this technique is that, to determine if a natural number p is prime, it is not necessary to know the prime numbers less than or equal to the square root of p. The objective of this work was to reduce the number of possibilities assumed by the integer variables <img src=image/13422919_04.gif> in the equation (i), (ii), (iii) respectively. Although it is true that this objective was achieved, we believe that the lower limits for the sums of the solutions of equations (i), (ii), (iii), were not optimal, since in our recent research we have managed to obtain limits lower, which reduce the domain of the integer variables <img src=image/13422919_04.gif> solve equations (i), (ii), (iii), respectively. In a future article we will show the results obtained. The methodology used was deductive and inductive. We would have liked to have a supercomputer, to build or determine prime numbers of many millions of digits, but this is not possible, since we do not have the support of our respective authorities. We believe that the contribution of this work to number theory is the creation of linear functionals for the study of integer solutions of quadratic polynomials in two variables, which represent a given natural number. The utility of large prime numbers can be used to encode any type of information safely, and the scheme shown in this article could be useful for this process.