Some results on number theory and differential equations

Abstract

In this work, using the basic tools of functional analysis, we obtain a technique that allows us to obtain important results, related to quadratic equations in two variables that represent a natural number and differential equations. We show the possible ways to write an even number that ends in six, as the sum of two odd numbers and we establish conditions for said odd numbers to be prime, also making use of a suitable linear functional F: R3 → R we obtain representations of natural numbers of the form (10A + 9), A ∈ N in order to obtain positive integer solutions of the equation quadratic (10x + 9)(10y + 9) = P where P is a natural number given that it ends with one. And finally, we show with three examples the use of the proposed technique to solve some ordinary and partial linear differential equations.We believe that the third corollary of our first result of this investigation can help to demonstrate the strong Goldbach conjecture.