Related papers: Primes in the intervals between primes squared

On the assumption of the Riemann hypothesis, we give explicit upper bounds on the difference between consecutive prime numbers.

We continue investigations on the average number of representations of a large positive integer as a sum of given powers of prime numbers. The average is taken over a short interval, whose admissible length depends on whether or not we…

We use short divisor sums to approximate prime tuples and moments for primes in short intervals. By connecting these results to classical moment problems we are able to prove that a positive proportion of consecutive primes are within a…

Let $p_{r+1}-1>n \geq p_r-1$, based on a sequence $\{1,2,3\cdots\ M_r(M_r=p_1p_2\cdots p_r)\}$, we compare the density of coprime numbers and establish a correlation between the proportions of coprime numbers in the ranges from 1 to…

Assuming the Riemann Hypothesis we prove that the interval $[N, N + H]$ contains an integer which is a sum of a prime and two squares of primes provided that $H \ge C (\log N)^{4}$, where $C > 0$ is an effective constant.

Let $X$ be a large parameter. We will first give a new estimate for the integral moments of primes in short intervals of the type $(p,p+h]$, where $p\leq X$ is a prime number and $h=\odi{X}$. Then we will apply this to prove that for every…

Let $k\ge 1$ be an integer. We prove that a suitable asymptotic formula for the average number of representations of integers $n=p_{1}^{k}+p_{2}^{2}+p_{3}^{2}$, where $p_1,p_2,p_3$ are prime numbers, holds in intervals shorter than the ones…

We analyze the inequality $\sqrt{P_{k+1}}-\sqrt{P_{k}}<1,\ k\in\mathbb{N}$, discuss the existence of primes on arbitrary intervals $(r,s),\ r<s,\ r,s\in\mathbb{R}$, and finally address the issue of primes between squares of naturals.

The variance of primes in short intervals relates to the Riemann Hypothesis, Montgomery's Pair Correlation Conjecture and the Hardy--Littlewood Conjecture. In regards to its asymptotics, very little is known unconditionally. We study the…

Define a natural number $n$ as a \textit{square-full} integer if for every prime $p$ such that $p|n$, we have $p^2|n$. In this paper, we establish an upper bound on the variance of square-full integers in short intervals of an expected…

In this work we show that based on a conjecture for the pair correlation of integers representable as sums of two squares, which was first suggested by Connors and Keating and reformulated here, the second moment of the distribution of the…

Assuming a $q$-variant of the prime $k$-tuple conjecture uniformly, we compute mixed moments of the number of primes in disjoint short intervals and progressions, respectively. This involves estimating the mean of singular series along…

We study values of k for which the interval (kn,(k+1)n) contains a prime for every n>1. We prove that the list of such integers k includes k=1,2,3,5,9,14, and no others, at least for k<=50,000,000. For every known k of this list, we give a…

In this paper, we show some results about the gap between a prime number and its consecutive prime number for large enough prime numbers. We show that the gap between a prime number $p_n$ and its consecutive prime number is not larger than…

We posit that $d_n^2 < 2p_{n+1}$ holds for all $n\geq 1$, where $p_n$ represents the $n$th prime and $d_n$ stands for the $n$th prime gap i.e. $d_n := p_{n+1} - p_n$. Then, the presence of a prime between successive perfect squares, as well…

We present and analyze an algorithm to enumerate all integers $n\le x$ that can be written as the sum of consecutive $k$th powers of primes, for $k>1$. We show that the number of such integers $n$ is asymptotically bounded by a constant…

The Prime Number Theorem states that the number of primes in $\{1,\ldots,x\}$, denoted $\pi(x)$, is approximately $\frac{x}{\ln(x)}$. In this paper, we investigate the distribution of primes for domains other than $\N$. First we look at…

In this paper we establish a number of new estimates concerning the prime counting function \pi(x), which improve the estimates proved in the literature. As an application, we deduce a new result concerning the existence of prime numbers in…

Let $g(n)$ be the largest positive integer $k$ such that there are distinct primes $p_i$ for $1\leq i\leq k$ so that $p_i |n+i$. This function is related to a celebrated conjecture of C.A. Grimm. We establish upper and lower bounds for…

We prove that suitable asymptotic formulae in short intervals hold for the problems of representing an integer as a sum of a prime square and a square, or a prime square. Such results are obtained both assuming the Riemann Hypothesis and in…