Some Results on Theory of Numbers, Partial Differential Equations and Numerical Analysis

Abstract

In this article, given a number <img src=image/13427314_01.gif> that ends in one and assuming that there are integer solutions <img src=image/13427314_02.gif> for the equations <img src=image/13427314_03.gif> or <img src=image/13427314_04.gif> or <img src=image/13427314_05.gif>, the straight line was used passing through the center of gravity of the triangle bounded by the vertices <img src=image/13427314_06.gif>. Considering A ≥ 25, we manage to divide the domain of the curve <img src=image/13427314_03.gif> into two disjoint subsets, and using Theorem (2.2) of this article, we find the subset where the integer solution of the equation <img src=image/13427314_03.gif> is found. Similar process is done when <img src=image/13427314_05.gif>, in case P is of the form <img src=image/13427314_04.gif> or <img src=image/13427314_07.gif>. These curves are different and to obtain a process similar to the one carried out previously, we proceeded according to Observation 2.2. Our results allow minimizing the number of operations to perform when our problem requires to be implemented computationally. Furthermore, we obtain some conditions to find the solution of the equations: <img src=image/13427314_08.gif> <img src=image/13427314_09.gif>, where <img src=image/13427314_10.gif> is of class <img src=image/13427314_11.gif>, <img src=image/13427314_12.gif> and <img src=image/13427314_13.gif> is a bounded open domain of <img src=image/13427314_14.gif> with piecewise smooth boundary <img src=image/13427314_15.gif>. All the operations carried out to find the solution have been carried out assuming that these exist, and we have found the conditions that <img src=image/13427314_16.gif> must satisfy for the coefficients <img src=image/13427314_17.gif>. We finish by finding an optimal domain for the real solution of a given polynomial of degree five. This process carried out on said given polynomial can also be carried out to reduce the degree of a given polynomial and thus obtain information about its roots.