相关论文: On the Ribenboim-Landau Conjecture

In this paper, we prove the twin prime conjecture showing that \begin{align} \sum \limits_{\substack{p\leq x\\p,p+2\in \mathbb{P}}}1\geq (1+o(1))\frac{x}{2\mathcal{C}\log^2 x}\nonumber \end{align} where $\mathcal{C}:=\mathcal{C}(2)>0$ fixed…

Let $k\geq 2$ be a fixed natural number. We establish the existence of infinitely many pairs of consecutive primes $p_n$, $p_{n+1}$ satisfying $$ p_{n+1}-p_n\geq c\:\frac{\log p_n\: \log_2 p_n\: \log_4 p_n}{\log_3 p_n}\:,$$ with $c$ being a…

For n>1, let G(n)=\sigma(n)/(n log log n), where \sigma(n) is the sum of the divisors of n. We prove that the Riemann Hypothesis is true if and only if 4 is the only composite number N satisfying G(N) \ge \max(G(N/p),G(aN)), for all prime…

Inspired by the proof of the Bertrand postulate given by P. Erd\H{o}S, we carefully examine and solve one less usual inequality in positive integers which could help to find an arithmetically pure proof that for every positive integer…

We study additive properties of consecutive prime numbers and the primality of the sums they generate. For a given prime number $p_n$, we consider the sums \[ S_k(p_n) = p_n + p_{n+1} + \cdots + p_{n+k-1}, \] where $k \ge 3$ is an odd…

A study of certain Hamiltonian systems has lead Y. Long to conjecture the existence of infinitely many primes of the form $p=2[\alpha n]+1$, where $1<\alpha<2$ is a fixed irrational number. An argument of P. Ribenboim coupled with classical…

Let $p_n$ be $n$th prime, and let $(S_n)_{n=1}^\infty:=(S_n)$ be the sequence of the sums of the first $2n$ consecutive primes, that is, $S_n=\sum_{k=1}^{2n}p_k$ with $n=1,2,\ldots$. Heuristic arguments supported by the corresponding…

In this note, we obtain a classification result for positive solutions to the critical p-Laplace equation in $\mathbb{R}^n$ with $n\ge4$ and $p>p_n$ for some number $p_n\in\left(\frac{n}{3},\frac{n+1}{3}\right)$ such that…

This paper introduces prime holdout problems, a problem class related to the Collatz conjecture. After applying a linear function, instead of removing a finite set of prime factors, a holdout problem specifies a set of primes to be…

It is well-known that for any non-constant polynomial $P$ with integer coefficients the sequence $(P(n))_{ n\in \mathbb N}$ has the property that there are infinitely many prime numbers dividing at least one term of this sequence.…

It is the purpose of this thesis to enunciate and prove a collection of explicit results in the theory of prime numbers. First, the problem of primes in short intervals is considered. We prove that there is a prime between consecutive cubes…

This paper introduces a new method to find the next prime number after a given prime ${P}$. The proposed method is used to derive a system of inequalities, that serve as constraints which should be satisfied by all primes whose successor is…

A well-known conjecture asserts that, for any given positive real number $\lambda$ and nonnegative integer $m$, the proportion of positive integers $n \le x$ for which the interval $(n,n + \lambda\log n]$ contains exactly $m$ primes is…

We prove that there is a prime between $n^3$ and $(n+1)^3$ for all $n \geq \exp(\exp(33.217))$. Our new tool which we derive is a version of Landau's explicit formula for the Riemann zeta-function with explicit bounds on the error term. We…

Let $p_n$ denote the $n$th prime and $g_n:=p_{n+1}-p_n$ the $n$th prime gap. We demonstrate the existence of infinitely many values of $n$ for which $g_n>g_{n+1}>\cdots>g_{n+m}$ with $m\gg \log\log\log n$ and similarly for the reversed…

Under Cram\'er's conjecture concerning the prime numbers, we prove that for any $x>1$, there exists a real $A=A(x)>1$ for which the formula $[A^{n^x}]$ (where $[]$ denotes the integer part) gives a prime number for any positive integer $n$.…

We prove that there are infinitely many solutions of $$ |\lambda_0+\lambda_1p+\lambda_2P_r|<p^{-\tau}, $$ where $r=3,$ $\tau=\frac1{118}$, and $\lambda_0$ is an arbitrary real number and $\lambda_1,\lambda_2\in\BR$ with $\lambda_2\neq0$ and…

We prove that given $\lambda \in \mathbb{R}$ such that $0 < \lambda < 1$, then $\pi(x + x^\lambda) - \pi(x) \sim \displaystyle \frac{x^\lambda}{\log(x)}$. This solves a long-standing problem concerning the existence of primes in short…

Let $P^+(n)$ denote the largest prime factor of the integer $n$ and $P_y^+(n)$ denote the largest prime factor $p$ of $n$ which satisfies $p\leqslant y$. In this paper, firstly we show that the triple consecutive integers with the two…

In this paper we provide the classification of positive solutions to the critical $p-$Laplace equation on $\mathbb{R}^n$, for $1<p<n$, possibly having infinite energy. If $n=2$, or if $n=3$ and $\frac 32<p<2$ we prove rigidity without any…