相关论文: Second numbers in arithmetic progressions
In this paper we study elliptic curves which have a number of points whose coordinates are in arithmetic progression. We first motivate this diophantine problem, prove some results, provide a number of interesting examples and, finally…
An efficient algorithm for classification of binary self-dual codes is presented. As an application, a complete classification of the self-dual codes of length 38 is given.
Our main result is that any real cubic algebraic number has a continued fraction expansion with polynomial coefficients. Some generalizations are mentioned.
We first introduce the Hamming distance between two strings. Then, we apply this concept to the representations of whole numbers in base n for all positive integers n > 2. We claim that a simple formula exists for the sum of all Hamming…
Partially or totally unsolved questions in number theory and geometry especially, such as coloration problems, elementary geometric conjectures, partitions, generalized periods of a number, length of a generalized period, arithmetic and…
We prove that there is a small but fixed positive integer e such that for every prime larger than a fixed integer, every subset S of the integers modulo p which satisfies |2S|<(2+e)|S| and 2(|2S|)-2|S|+2 < p is contained in an arithmetic…
This is an article for a general mathematical audience on the author's work, joint with Terence Tao, establishing that there are arbitrarily long arithmetic progressions of primes. It is based on several one hour lectures, chiefly given at…
In this paper, we introduce the degenerate central factorial polynomials and numbers of the second kind which are degenerate versions of the central factorial polynomials and numbers of the second kind. We derive some properties and…
A complete classification of binary doubly even self-dual codes of length 40 is given. As a consequence, a classification of binary extremal self-dual codes of length 38 is also given.
We show that there exists a bounded pattern of m consecutive primes for any m>0, that means a tuple H_m of m distinct non-negative integers h_i (i=1,2,...m) such that its translations contain arbitrarily long (finite) arithmetic…
We show that there exists a positive constant C such that the following holds: Given an infinite arithmetic progression A of real numbers and a sufficiently large integer n (depending on A), there needs at least Cn geometric progressions to…
In this project we show the existence of arbitrary length arithmetic progressions in model sets and Meyer sets in the Euclidean $d$-space. We prove a van der Waerden type theorem for Meyer sets. We show that pure point subsets of Meyer sets…
Let $\alpha_1, \cdots, \alpha_d$ be real numbers, and let $S$ be the set of integers $s$ so that $||\alpha_i s||_{\mathbb{R}/\mathbb{Z}}>\delta$ for some $i$ and some fixed $\delta>0$. We prove $S$ is not \enquote{$2$-large}, i.e. there is…
The aim of the present article is to explore the possibilities of representing positive integers as sums of other positive integers and highlight certain fundamental connections between their multiplicative and additive properties. In…
By using the strong approximation, this paper establishes several limit results on the convergent rate of a infinite series of probabilities on the other law of iterated logarithm.
Suppose that $\xi^{(n)}_1,\xi^{(n)}_2,...,\xi^{(n)}_n$ are i.i.d with $P(\xi^{(n)}_i=1)=p_n=1-P(\xi^{(n)}_i=0)$. Let $U^{(n)}$ and $W^{(n)}$ be the longest length of arithmetic progressions and of arithmetic progressions mod $n$ relative to…
In this paper we study a sequence involving the prime numbers by deriving two asymptotic formulas and finding new upper and lower bounds, which improve the currently known estimates.
We calculate the formal analytic expansions of certain formal translations in a space of formal iterated logarithmic and exponential variables. The results show how the algebraic structure naturally involves the Stirling numbers of the…
Some symmetry problems are formulated and solved. New simple proofs are given for the earlier studied symmetry problems.
We introduce a new permutation statistic, namely, the number of cycles of length $q$ consisting of consecutive integers, and consider the distribution of this statistic among the permutations of $\{1,2,...,n\}$. We determine explicit…