Related papers: Gaps between Primes in Beatty Sequences

There are several centrality measures that have been introduced and studied for real world networks. They account for the different vertex characteristics that permit them to be ranked in order of importance in the network. Betweenness…

We study pattern densities in binary sequences, finding optimal limit sequences with fixed pattern densities.

Let us call a simple graph on $n\geq 2$ vertices a prime gap graph if its vertex degrees are $1$ and the first $n-1$ prime gaps. We show that such a graph exists for every large $n$, and in fact for every $n\geq 2$ if we assume the Riemann…

I examine the phenomenology of particle multiplicity distributions, with special emphasis on the low multiplicities that are a background in the study of rapidity gaps. In particular, I analyze the multiplicity distribution in a rapidity…

For each $m\geq 1$, there exist infinitely many primes $p_1<p_2<\ldots<p_{m+1}$ such that $p_{m+1}-p_1=O(m^4e^{8m})$ and $p_j+2$ has at most $\frac{16m}{\log 2}+\frac{5\log m}{\log 2}+37$ prime divisors for each $j$.

In this paper we study the problem of detecting prime numbers between all consecutive cubes. Firstly, we use a large computation to show that there is always a prime between $n^3$ and $(n+1)^3$ for $n^3\leq 1.649\cdot 10^{40}$. In addition,…

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…

We study the spacing of the primes using methods from information theory. In information theory, the equivalence of continuous and discrete representations of information is established by Shannon sampling theory. Here, we use Shannon…

Bertrand's postulate establishes that for all positive integers $n>1$ there exists a prime number between $n$ and $2n$. We consider a generalization of this theorem as: for integers $n\geq k\geq 2$ is there a prime number between $kn$ and…

This paper presents a distinctive prime detection approach. This method use GM-(n+1) sequences to effectively eliminate complex numbers. The sequences, which consist of odd a number of (n+1), exclude all components except for the initial…

In this paper, using the well known fact that the series of reciprocals of primes diverges, we obtain a general inequality for gaps of consecutive primes that holds for infinitely many primes. As it is shown the key ingredient for this…

In 2016 Lemke Oliver and Soundararajan examined the gaps between the first hundred million primes and observed biases in their distributions modulo 10. Given our work on the evolution of the populations of various gaps across stages of…

As a refinement of the celebrated recent work of Yitang Zhang we show that any admissible k-tuple of integers contains at least two primes and almost primes in each component infinitely often if k is at least 181000. This implies that there…

In the present paper, we have developed a method for solving \textit{diophantine inequalities} using their relationship with the \textit{difference between consecutive primes}. Using this approach we have been able to prove some theorems,…

We consider string matching with variable length gaps. Given a string $T$ and a pattern $P$ consisting of strings separated by variable length gaps (arbitrary strings of length in a specified range), the problem is to find all ending…

We resolve a function field version of two conjectures concerning the variance of the number of primes in short intervals (Goldston and Montgomery) and in arithmetic progressions (Hooley). A crucial ingredient in our work are recent…

We call an interval $[x,y]$ in a poset {\em small} if $y$ is the join of some elements covering $x$. In this paper, we study the chains of paths from a given arbitrary (binary) path $P$ to the maximum path having only small intervals. More…

74 new integer sequences are introduced in number theory, and for each of them is given a characterization, followed by open problems. each one a general question: how many primes each sequence has.

We propose a multi-scale analysis method for studying arithmetic properties of integer sets, such as primality. Our approach organizes information through a hierarchy of nested sequences, where each level enables a hierarchical expression…

Let $\alpha,\beta$ be real numbers such that $\alpha>1$ is irrational and of finite type, and let $c$ be a real number in the range $1<c<\frac{14}{13}$. In this paper, it is shown that there are infinitely many Piatetski-Shapiro primes $p =…