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 : R 3 → 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.

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:

No abstract