Some Results on Number Theory and Differential Equations

Maguiña, B. M. Cerna; Garcia, Dik D. Lujerio; Maguina, Hector F.

doi:10.13189/ms.2021.090614

Cited by 3 publications

(3 citation statements)

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“…Combining the results obtained in [1] and [10], via the Theorems (2.1), (2.2) using a line that passes through the center of gravity of the triangle bounded by the vertices (A, A), (B, A) and (A, B) we manage to divide the domain of the equation P = (10x + 9)(10y + 9) into two disjoint subsets and via Theorem (2.2), we determine in which of the subsets the integer solution lies. We show this process with an example.…”

Section: Introductionsupporting

confidence: 62%

“…Using the relation (3), the relation (10) and the data P a ≤ P 1/2 259 we get one of the desired results. For 1 < P a < 2, we see that the line L does not intersect the equation (1) and using the relation (3) we obtain the mentioned results.…”

Section: On Results In Number Theorymentioning

confidence: 93%

“…In the articles [1] and [10] some results are obtained on integer solutions of quadratic polynomials in two variables. In Theorem (2.1) of [1] it is obtained that the face of the integer solution of the equation P = (10x + 9)(10y + 9) is given by where N and M are relatively prime and P is a natural number ending in one.…”

Section: Introductionmentioning

confidence: 99%

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

Maguiña¹,

Garcia²,

Pareja³

et al. 2022

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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:

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Section: Introductionsupporting

confidence: 62%

Section: On Results In Number Theorymentioning

confidence: 93%

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

Maguiña¹,

Garcia²,

Pareja³

et al. 2022

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New Perspectives on the Roots of Real Polynomials of Degree n and Number Theory

Maguiña,

Lujerio Garcia,

Jauregui

et al. 2024

Preprint

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In this work we obtained some results on the real and complex roots of real polynomials of the form \(P(x)=x^{n}+a_{n-1}x^{n-1}+\dots +a_{1 }x+a_{0}\), where the \(a_{k}\) are real numbers for \(k\in \{0,\dots ,n-1 \}\), we also obtained results on linear Diophantine equations of the form \(ax+by+cz=d\), where \(a,b,c\) and \(d\) are integers. To obtain the desired results, the relation that states the following was used: for any real numbers \(A\) and \(B\), there exists a unique real number \(\lambda\), such that \(A+B=\lambda [A^2+ B^2]\). This result is appropriately linked to the object of study. For the polynomials, optimal domains were obtained where the real or complex roots are found, without the use of higher calculus. For the linear Diophantine equation, the desired solutions were obtained, which were found by establishing several links between the Diophantine equation and the relationship \(A+B=\lambda [A^2+B^2]\). Several examples of the results obtained are illustrated, which are intended to show the benefits of this proposal. Furthermore, using the same technique used to solve the previous problems, we propose an alternative method to address Fermat’s Last Theorem.

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Some Results on Maxima and Minima of Real Functions of Vector Variables: A New Perspective

Maguiña,

Lujerio Garcia,

Pocoy Yauri

et al. 2024

Preprint

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In this work, the extreme points of real vector variable functions are obtained without the use of the classical theory that involves the use of partial derivatives. We illustrate with several theorems and examples a new method that consists of establishing an appropriate link between the function to be optimized, its restrictions and the result, stating that: given \(n\) non-zero real numbers \(a_{1},a_{2},\cdots,a_{n}\in\mathbb{R}\), then there exists a unique \(\lambda\in\mathbb{R}\) such that: This relation is obtained by decomposing the Hilbert space \(\mathbb{R}^{n}\) as the direct sum of a closed subspace and its orthogonal complement. Since the dimension of the space \(\mathbb{R}^{n}\) is finite, this guarantees that any linear functional defined on the space \(\mathbb{R}^{n}\) is continuous, and this guarantees that the kernel of said linear functional is closed in the space \(\mathbb{R}^{n}\), therefore we have that the space \(\mathbb{R}^{n}\) breaks down, as the direct sum of the kernel of the continuous linear functional \(f\) and its orthogonal complement, that is: \(\mathbb{R}^{n}\,-\,\ker f\,\bigoplus\,\left[\,\ker f\,\right]^{\perp}\), where the dimension of \(\ker f\,-\,n\,-\,1\) and the dimension of \(\left[\,\ker f\,\right]^{\perp}\,-\,1\). Adding to the link found new definitions about the hierarchy of one variable in relation to another and the fact that if \(x_{1}^{2}+x_{2}^{2}+...+x_{n}^{2}\,-\,r^{2}\) then the \(\max\{x_{1}+x_{2}+\cdots+x_{n}\}\,-\,r\sqrt{n}\) and the \(\min\{x_{1}+x_{2}+\cdots+x_{n}\}\,-\,-r\sqrt{n}\) we solve the optimization problem without using classical theory.

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Some Results on Number Theory and Differential Equations

Cited by 3 publications

References 1 publication

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

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

New Perspectives on the Roots of Real Polynomials of Degree n and Number Theory

Some Results on Maxima and Minima of Real Functions of Vector Variables: A New Perspective

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