Mathematics and Statistics 10(2): 442-453, 2022
DOI: 10.13189/ms.2022.100220
http://www.hrpub.org
Some Results on Number Theory and Analysis
B. M. Cerna Magui ˜ na
1
, Dik D. Lujerio Garcia
1,*
, H´ ector F. Magui˜ na
2
, Miguel A. Tarazona Girlado
3
1
Academic Department of Mathematics, Science Faculty, Santiago Ant´ unez de Mayolo National University, Shancayan Campus, Per´ u
2
National University of San Marcos, Street Germ´ an Am´ ezaga N°375. Pabell´ on ”F”-Accounting Sciences University City, Per´ u
3
Faculty of Electronic and Computer Engineering, Federico Villarreal National University,Per´ u
Received February 18, 2021; Revised March 7, 2022; Accepted March 27, 2022
Cite This Paper in the following Citation Styles
(a): [1] B. M. Cerna Magui˜ na, Dik D. Lujerio Garcia, H´ ector F. Magui˜ na, Miguel A. Tarazona Girlado, ”Some Results on Number Theory and Analysis,”
Mathematics and Statistics, Vol.10, No.2, pp. 442-453, 2022. DOI: 10.13189/ms.2022.100220
(b): B. M. Cerna Magui˜ na, Dik D. Lujerio Garcia, H´ ector F. Magui˜ na, Miguel A. Tarazona Girlado,, (2022). Some Results on Number Theory and Analysis.
Mathematics and Statistics, 10(2), 442-453 DOI: 10.13189/ms.2022.100220
Copyright ©2022 by authors, all rights reserved. Authors agree that this article remains permanently open access under the terms of
the Creative Commons Attribution License 4.0 International License
In memory of Emiliana Magui ˜ na Cabana.
Abstract In this work we obtain bounds for the sum of the integer solutions of quadratic polynomials of two variables of the
form P = (10x + 9)(10y + 9) or P = (10x + 1)(10y + 1) or P = (10x + 7)(10y + 3) where P is a given natural number
that ends in one. This allows us to decide the primality of a natural number P that ends in one. Also we get some results on
twin prime numbers. In addition, we use special linear functionals defined on a real Hilbert space of dimension n, n ≥ 2 , in
which the relation is obtained: a
1
+ a
2
+ ··· + a
n
= λ[a
2
1
+ ··· + a
2
n
], where a
i
is a real number for i =1, ..., n. When
n =3 or n =2 we manage to address Fermat’s Last Theorem and the equation x
4
+ y
4
= z
4
, proving that both equations do not
have positive integer solutions. For n =2, the Cauchy-Schwartz Theorem and Young’s inequality were proved in an original way.
Keywords Diophantine Equation, Prime Numbers, Twin Prime Numbers, Cauchy-Shwarz Inequality, Fermat’s Last Theorem
1 Introduction
We know that for a Hilbert space H and M a closed subspace of H we have that H = M ⊕ M
⊥
, following the ideas given
in the article [5] and [6] we create linear functionals f : R
n
→ R obtaining the relation:
a
1
+ a
2
+ ··· + a
n
= λ(a
2
1
+ a
2
2
+ ··· + a
2
n
)
where the a
i
∈ R. We have only used the cases n =2 or n =3, or the combination of both cases.
In the study of quadratic polynomials in two variables that represent a natural number, better bounds are obtained for the sum
A + B, where (A, B) is the integer solution of one of the equations P = (10x + 9)(10y + 9) or P = (10x + 1)(10y + 1) or
P = (10x + 7)(10y + 3), where P is a natural number that ends in one.
By studying Fermat’s equation, x
n
+ y
n
= z
n
and assuming that there is an integer solution for n ≥ 3, we obtained the
equation of a sphere, that is :
a
1
−
1
2λ
2
+
a
2
−
1
2λ
2
+
a
3
−
1
2λ
2
=
3
4λ
2
.