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

Cerna Maguiña B.M.; Garcia D.D.L.; Pareja C.R.; Cinthia T.D. · Mathematics and Statistics (2022) · DOI: 10.13189/ms.2022.100512

Resumen

In this article, given a number P ∈ N that ends in one and assuming that there are integer solutions (A; B) ∈ N × N for the equations P = (10x + 9)(10y + 9) or P = (10x + 7)(10y + 3) or P = (10x + 1)(10y + 1), the straight line was used passing through the center of gravity of the triangle bounded by the vertices (A; A), (B; A), (A; B). Considering A ≥ 25, we manage to divide the domain of the curve P = (10x + 9)(10y + 9) into two disjoint subsets, and using Theorem (2.2) of this article, we find the subset where the integer solution of the equation P = (10x + 9)(10y + 9) is found. Similar process is done when P = (10x + 1)(10y + 1), in case P is of the form P = (10x + 7)(10y + 3) or P = (10x + 3)(10y + 7). 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: (Formula presented) where u is of class C2, n = 2 and D is a bounded open domain of R2 with piecewise smooth boundary ∂D. All the operations carried out to find the solution have been carried out assuming that these exist, and we have found the conditions that f must satisfy for the coefficients aij . We finish by finding an optimal domain for the real solution of a given polynomial of degree five. This process carriedout on said given polynomial can also be carried out to reduce the degree of a given polynomial and thus obtain information about its roots.

Instituciones: Universidad Nacional Santiago Antúnez de Mayolo