On Primes In Short Intervals

On a new method for calculation of the number of prime numbers in the given interval

Definition of the number of prime numbers in the given interval

单位区间中素数不平衡性的密度研究

On the Density of Prime Imbalances in the Unit Interval

本研究证明素数归一化差值集合在开单位区间稠密，提出带量化界的显式构造算法，基于Bertrand假设等素数理论基础结果完成证明。

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.

On a Deterministic Property of the Category of $k$-almost Primes: A Deterministic Structure Based on a Linear Function for Redefining the $k$-almost Primes ($\exists n\in {\rm N} $, $1{\le} k {\le}n$) in Certain Intervals

In this paper based on a sort of linear function, a deterministic and simple algorithm with an algebraic structure is presented for calculating all (and only) $k$-almost primes ($where$ $\exists n\in {\rm N} $, $1{\le} k {\le}n$) in certain interval. A theorem has been proven showing a new deterministic property of the category of $k$-almost primes. Through a linear function that we obtain, an equivalent redefinition of the $k$-almost primes with an algebraic characteristic is identified. Moreover, as an outcome of our function's property some relations which contain new information about the …

On the existence of twin prime in an interval

Let $S_{(x,y]} = \left\{\frac{p_n}{p_{n+1}-2} :~ n\in I \right\}$, where $I = \left\{n :~ x<p_n \le y \right\}$, $p_n$ is the $n$-th prime and $x, y \in \mathbb{R}_{>0}$. If $M_α(x,y)$ denotes the $α$-power mean of the elements of $S_{(x,y]}$, it is shown that the existence of a twin prime pair in $(x,y]$ is implied if $\displaystyle \lim_{α\rightarrow \infty}M_α(x,y) > 1 - 2/y + O(y^{-2})$ for a sufficiently large $y$. For a special choice of $y$, we also find a lower bound for the mean: $\displaystyle \lim_{α\rightarrow \infty}M_α(x,x^β)>1-c/x^β+O(x^{-β}\log^{-1} x)$, where the constant $c>0…

Einige Sätze über Primzahlen und spezielle binomische Ausdrücke / [english] Some Theorems about prime numbers and Special Binomial expressions

1. There is no existing any quadratic interval $η_{n}:=(n^{2},(n+1)^{2}],$ which contains less than 2 prime numbers. The number of prime numbers within $η_{n}$ goes averagely linear with n to infinity. 2. The exact law of the number $π(n)$ of prime numbers smaller or equal to n is given. As an approximation of that we get the prime number theorem of Gauss for great values of n. 3. We derive partition laws for $π(η_{n})$, for the number of twin primes $π_{2}(η_{n})$ in quadratic intervals $η_{n}$ and for the multiplicity $π_{g}(2n)$ of representations of Goldbach-pairs for a given even number 2…