We speculate on the distribution of primes in exponentially growing, linear recurrence sequences (un)n≥0 in the integers. By tweaking a heuristic which is successfully used to predict the number of ...prime values of polynomials, we guess that either there are only finitely many primes un, or else there exists a constant cu>0 (which we can give good approximations to) such that there are ∼culogN primes un with n≤N, as N→∞. We compare our conjecture to the limited amount of data that we can compile. One new feature is that the primes in our Euler product are not taken in order of their size, but rather in order of the size of the period of the un(modp).
Análisis y Programación de los Números Primos Figueroa Molina, Roberto; Nieves Vanegas, Sergio; Romero Pabon, Julio Cesar
Prospectiva (Barranquilla. Online),
2021, Volume:
19, Issue:
2
Journal Article
Peer reviewed
Open access
The study of prime numbers is a subject of great importance for mathematics, because they are essential for
the fundamental pillars of Arithmetic, as is the case with its Fundamental Theorem, which ...states that any
number can be decomposed into a single product.
of prime numbers. This concept of decomposing a number
into unique factors was introduced by Euclid, who made great contributions to mathematics and geometry.
This work presents an algorithm to obtain the prime numbers in a highly considered set, as
well as its analysis related to the number of prime numbers that exist in a given number interval, their organization,
classification and differences that exist between them. Prime numbers are currently being studied as they are used to encode any type of
information in a secure way, since these numbers are unique and do not adhere to any rule or pattern to build them.
El estudio de los números primos es un tema esencial para las matemáticas, como el caso del Teorema Fundamental de la Aritmética, afirma que, cualquier número puede descomponerse en un producto único de números primos. El concepto de descomponer un número en factores únicos lo introdujo Euclides 1, quien hizo grandes aportes a las matemáticas y a la geometría. En este trabajo se presenta un algoritmo, para obtener los números primos de un conjunto grandemente estimado, como también el análisis relacionado con la cantidad de números primos que concurren en determinado intervalo de números, su organización, clasificación y diferencias que coexisten entre ellos. En la actualidad los números primos son altamente estudiados, se emplean para codificar cualquier tipo de información de forma segura, puesto que, estos números son únicos y no se ajustan a ninguna regla o patrón para construirlos.
Let
a
be an ideal of Noetherian ring R and M a finitely generated R-module such that
cd
(
a
,
M
)
=
c
. In this paper, we investigate
Att
R
(
H
a
c
(
M
)
)
. Among other things, it is shown that
Max
...{
p
∈
Supp
R
M
|
cd
(
a
,
R
/
p
)
=
c
}
⊆
Att
R
(
H
a
c
(
M
)
)
. We also show that
Att
R
(
H
a
c
(
M
)
)
=
{
p
∈
Supp
R
M
|
cd
(
a
,
R
/
p
)
=
c
,
p
=
Ann
R
(
H
a
c
(
R
/
p
)
)
}
and
{
p
∈
Supp
R
M
|
cd
(
a
,
R
/
p
)
=
c
,
dim
R
/
p
−
1
≤
cd
(
a
,
R
/
p
)
≤
dim
R
/
p
}
⊆
Att
R
(
H
a
c
(
M
)
)
.
Finally, we prove that if
(
R
,
m
)
is a local ring and
dim
R
/
a
=
1
then
Att
R
(
H
a
c
(
M
)
)
=
{
p
∈
Supp
R
M
|
cd
(
a
,
R
/
p
)
=
cd
(
a
,
M
)
}
. Then by using this, it is shown that if
(
R
,
m
)
is a local ring then
{
p
∈
Supp
R
M
|
cd
(
a
,
R
/
p
)
=
c
,
dim
R
/
(
a
+
p
)
=
1
}
⊆
Att
R
(
H
a
c
(
M
)
)
.
Purpose
The distribution of natural numbers in the Ulam spiral manifests a series of unexpected regularities of the elusive prime numbers. Their sequencing remains a topic of research interest, with ...many ramifications in different branches of Mathematics, especially in number theory and the prime factorisation problem. Accordingly, the focus of the research is on the most known and widespread asymmetric cryptographic system, that is, the RSA encryption.
Design/methodology/approach
This paper presents the presence of one, two, three or four adjacencies for the diverse primes that appear in a spiral, considering the Hardy–Littlewood twin prime conjecture and the constellations of primes.
Findings
Through equations, the calculation formulas of primes inside a spiral that have one to four primes in their adjacent places is offered, with approximate expressions that facilitate the operations, showing that the adjacencies decrease very rapidly as the spiral progresses, although without disappearing.
Originality/value
A generalisation to cases in which the distances to the prime values change in an ascending way in accordance with the step of the Ulam spiral is offered.
The Bateman–Horn conjecture is a far-reaching statement about the distribution of the prime numbers. It implies many known results, such as the prime number theorem and the Green–Tao theorem, along ...with many famous conjectures, such the twin prime conjecture and Landau’s conjecture. We discuss the Bateman–Horn conjecture, its applications, and its origins.
Let r1,…,rt be positive integers and let R2(r1,…,rt) be the set of positive odd integers that can be represented as p+2k1r1+⋯+2ktrt, where p is a prime and k1,…,kt are positive integers. It is easy ...to see that if r1−1+⋯+rt−1<1, then the set R2(r1,…,rt) has asymptotic density zero. In this paper, we prove that if r1−1+⋯+rt−1≥1, then the set R2(r1,…,rt) has a positive lower asymptotic density. Several conjectures and open problems are posed for further research.
We prove an explicit error term for the \psi (x,\chi ) function assuming the Generalized Riemann Hypothesis. Using this estimate, we prove a conditional explicit bound for the number of primes in ...arithmetic progressions.
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