关于三个连续素数

最小连续原根对

Least Consecutive Pair of Primitive Roots

研究了大素数p中最小连续原根对u和v^2及u+1的上界性质，采用数学分析方法证明其与x的大小关系。

关于连续素数和的猜想

Conjectures on Sums of Consecutive Primes

研究连续素数和的素数性，提出存在性和无限性猜想，并通过概率和模数分析支持这些猜想。

关于素数之间的最大间距

On the Maximal Gap between Primes

本文证明了对于足够大的素数，其与相邻素数的间距不超过$2\log^2{p_n}$，并由此推导出特定区间内存在素数。

最小连续二次非剩余对

Least Consecutive Pair of Quadratic Nonresidues

研究大素数域中最小连续二次非剩余对的上界，证明其不超过 O((log p)^2 (log log p)^3)。

The importance of finding the upper bounds for prime gaps in order to solve the twin primes conjecture and the Goldbach conjecture

ABSTRACT. In this article we present a point of view that highlights the importance of finding the upper bounds for prime gaps, in order to solve the twin primes conjecture and the Goldbach conjecture. For this purpose, we present a procedure for the determination of the upper bounds for prime gaps different from the most famous and known approaches. The proposed method analyzes the distribution of prime numbers using the set of relative numbers. Using negative numbers too, it becomes intuitive to understand that that the arrangement of 2P+1 consecutive numbers that goes -P to P, is the only a…

连续素数分布中的预期偏差

Expected biases in the distribution of consecutive primes

本文基于埃拉托斯特尼筛法各阶段的间隔种群演化，解释了Lemke Oliver和Soundararajan观察到的素数间隔模10分布偏差，指出该偏差源于小间隔的快速起步和剩余类分配不均，且最终是暂时的。

使用改进的拉格朗日多项式生成与分布素数

Generation and Distribution of Prime Numbers Using a Modified Lagrange Polynomial

本文提出一种改进的拉格朗日多项式外推方法，并基于连续素数间的线性递推关系设计素数生成算法，用于从已知连续素数生成新的素数。

Consecutive Quadratic Residues And Quadratic Nonresidue Modulo $p$

Let $p$ be a large prime, and let $k\ll \log p$. A new proof of the existence of any pattern of $k$ consecutive quadratic residues and quadratic nonresidues is introduced in this note. Further, an application to the least quadratic nonresidues $n_p$ modulo $p$ shows that $n_p\ll (\log p)(\log \log p)$.

关于差距的说明

A note on gaps

令$p_{k}$表示第$k$个素数，$d(p_{k}) = p_{k} - p_{k - 1}$表示连续素数之间的差。我们用$N_ε(x)$表示满足不等式$d(p_{k}) \leq (\log p_{k})^{2 + ε}$的素数$\leq x$的个数，其中$ε> 0$是任意的和固定的，用$π(x)$表示小于或等于$x$的素数的个数。本文首先证明了一个定理，即$\lim_{x \to \infty} N_ε(x)/π(x) = 1$ .一个推论的定理证明之间的差距连续squarefree数的陈述。

Configurations Of Consecutive Primitive Roots

Let $p \geq 2$ be a large prime, and let $k \ll \log p $ be a small integer. This note proves the existence of various configurations of $(k+1)$-tuples of consecutive and quasi consecutive primitive roots $n+a_0, n+a_1, n+a_2, \ldots, n+a_k$ in the finite field $\mathbb{F}_p$, where $a_0,a_1, \ldots, a_k$ is a fixed $(k+1)$-tuples of distinct integers.

Note On The Catalan Constant And Prime Triples

The existence of infinitely many consecutive prime triples $p_n$, $ p_{n+1}$, and $p_{n+2}$ as $n \to \infty$, is sufficient to prove that the Catalan constant $β(2)=0.9159655941\ldots $ is an irrational number. This note provides the detailed analysis. Moreover, the numerical data suggests that the irrationality measure is $μ(β(2))=2$, the same as almost every irrational real numbers.

On solutions of some of unsolved problems in number theory, specifically on the distribution of primes

We solve some famous conjectures on the distribution of primes. These conjectures are to be listed as Legendre's, Andrica's, Oppermann's, Brocard's, Cramér's, Shanks', and five Smarandache's conjectures. We make use of both Firoozbakht's conjecture (which recently proved by the author) and Kourbatov's theorem on the distribution of and gaps between consecutive primes. These latter conjecture and theorem play an essential role in our methods for proving these famous conjectures. In order to prove Shanks' conjecture, we make use of Panaitopol's asymptotic formula for $π(x)$ as well.

Asymptotic of the greatest distance between adjacent primes and the Hardy-Littlewood conjecture

The paper substantiates the conjecture of the asymptotic behavior of the largest distance between consecutive primes: $sup_{p_i \leq x}(p_{i+1}-p_i) \sim 2e^{-γ} \log^2(x)$, where $γ$ is the Euler constant. The Hardy-Littlewood conjecture about the number of prime tuples is investigated and the rationale for this conjecture is given, taking into account the fact that a large natural number is not divisible by primes. It also substantiates why the accuracy of this conjecture is not affected by another assumption about the probability of a natural number being prime, although such a probability …

推广或与Andrica猜想有关的六个猜想

Six Conjectures which Generalize or Are Related to Andrica's Conjecture

本文给出了六种连续素数对的证明，并举例说明。