A Refinement of the Function g(m)g(m) on Grimm Conjecture

In this paper, we refine the function $g(x)$ on Grimm's conjecture and improve a result of Erd\"{o}s and Selfridge without using Hall's theorem.Comment: The paper has been accepted by THE ROCKY MOUNTAIN JOURNAL OF MATHEMATICS, to appea

Euclid's Number-Theoretical Work

When people mention the mathematical achievements of Euclid, his geometrical achievements always spring to mind. But, his Number-Theoretical achievements (See Books 7, 8 and 9 in his magnum opus \emph{Elements} [1]) are rarely spoken. The object of this paper is to affirm the number-theoretical role of Euclid and the historical significance of Euclid's algorithm. It is known that almost all elementary number-theoretical texts begin with Division algorithm. However, Euclid did not do like this. He began his number-theoretical work by introducing his algorithm. We were quite surprised when we began to read the \emph{Elements} for the first time. Nevertheless, one can prove that Euclid's algorithm is essentially equivalent with the Bezout's equation and Division algorithm. Therefore, Euclid has preliminarily established Theory of Divisibility and the greatest common divisor. This is the foundation of Number Theory. After more than 2000 years, by creatively introducing the notion of congruence, Gauss published his \emph{Disquisitiones Arithmeticae} in 1801 and developed Number Theory as a systematic science. Note also that Euclid's algorithm implies Euclid's first theorem (which is the heart of `the uniqueness part' of the fundamental theorem of arithmetic) and Euclid's second theorem (which states that there are infinitely many primes). Thus, in the nature of things, Euclid's algorithm is the most important number-theoretical work of Euclid. For this reason, we further summarize briefly the influence of Euclid's algorithm. Knuth said `we might call Euclid's method the granddaddy of all algorithms'. Based on our discussion and analysis, it leads to the conclusion Euclid's algorithm is the greatest number-theoretical achievement of the Euclidean age.Comment: 32 pages; We give a detailed revisio

Dickson's conjecture on ZnZ^n--An equivalent form of Green-Tao's conjecture

In [1], we give Dickson's conjecture on $N^n$. In this paper, we further give Dickson's conjecture on $Z^n$ and obtain an equivalent form of Green-Tao's conjecture [2]. Based on our work, it is possible to establish a general theory that several multivariable integral polynomials on $Z^n$ represent simultaneously prime numbers for infinitely many integral points and generalize the analogy of Chinese Remainder Theorem in [3]. Dans [1], nous donnons la conjecture de Dickson sur $N^n$. Dans ce document, en outre nous accordons une conjecture de Dickson sur $Z^n$ et obtenons une forme \'{e}quivalent de conjecture de Green-Tao [2]. Sur la base de nos travaux, il est possible d'\'{e}tablir une th\'{e}orie g\'{e}n\'{e}rale que plusieurs polyn\^{o}mes int\'{e}graux multivariables sur $Z^n$ repr\'{e}sentent simultan\'{e}ment les nombres premiers pour un nombre infini de points entiers et de g\'{e}n\'{e}raliser les l'analogie de Th\'{e}or\`{e}me des Restes Chinois dans [3].Comment: 8 page

The problem of the least prime number in an arithmetic progression and its applications to Goldbach's conjecture

The problem of the least prime number in an arithmetic progression is one of the most important topics in Number Theory. In [11], we are the first to study the relations between this problem and Goldbach's conjecture. In this paper, we further consider its applications to Goldbach's conjecture and refine the result in [11]. Moreover, we also try to generalize the problem of the least prime number in an arithmetic progression and give an analogy of Goldbach's conjecture.Comment: Give a detailed proof of Theorem 1 and add Theorem 2 and Appendi

W Sequences and the Distribution of Primes in Short Interval

Based on Euclid's algorithm, we find a kind of special sequences which play an interesting role in the study of primes. We call them W Sequences. They not only ties up the distribution of primes in short interval but also enables us to give new weakened forms of many classical problems in Number Theory. The object of this paper is to provide a brief introduction and preliminary analysis on this kind of special sequences.Comment: The paper has been improved and shortened. The title is changed. Accepted by JP Journal of Algebra, Number Theory and Application

A new inequality involving primes

In this note, we find a new inequality involving primes and deduce several Bonse-type inequalities.Comment: 5 page

On the Infinitude of Some Special Kinds of Primes

The aim of this paper is to try to establish a generic model for the problem that several multivariable number-theoretic functions represent simultaneously primes for infinitely many integral points. More concretely, we introduced briefly the research background-the history and current situation-from Euclid's second theorem to Green-Tao theorem. We analyzed some equivalent necessary conditions that irreducible univariable polynomials with integral coefficients represent infinitely many primes, found new necessary conditions which perhaps imply that there are only finitely many Fermat primes, obtained an analogy of the Chinese Remainder Theorem, generalized Euler's function, the prime-counting function and Schinzel-Sierpinski's Conjecture and so on. Nevertheless, this is only a beginning and it miles to go. We hope that number theorists consider further it.Comment: Added recent results, see [101-103]. This paper is dedicated to my great mother Caiying Cao, born on January 12, 1943--died on August 10, 198

Revisiting the number of simple K4K_4-groups

In this paper, by solving Diophantine equations involving simple $K_4$-groups, we will try to point out that it is not easy to prove the infinitude of simple $K_4$-groups. This problem goes far beyond what is known about Dickson's conjecture at present.Comment: 9 page

Generalizations of an Ancient Greek Inequality about the Sequence of Primes

In this note, we generalize an ancient Greek inequality about the sequence of primes to the cases of arithmetic progressions even multivariable polynomials with integral coefficients. We also refine Bouniakowsky's conjecture [16] and Conjecture 2 in [22]. Moreover, we give two remarks on conjectures in [22]Comment: 8 page

A Matrix-in-matrix Neural Network for Image Super Resolution

In recent years, deep learning methods have achieved impressive results with higher peak signal-to-noise ratio in single image super-resolution (SISR) tasks by utilizing deeper layers. However, their application is quite limited since they require high computing power. In addition, most of the existing methods rarely take full advantage of the intermediate features which are helpful for restoration. To address these issues, we propose a moderate-size SISR net work named matrixed channel attention network (MCAN) by constructing a matrix ensemble of multi-connected channel attention blocks (MCAB). Several models of different sizes are released to meet various practical requirements. Conclusions can be drawn from our extensive benchmark experiments that the proposed models achieve better performance with much fewer multiply-adds and parameters. Our models will be made publicly available