Related papers: The ternary Goldbach problem with primes in positi…

We consider the ternary Goldbach problem with two prime variables of the form $k^2+m^2+1$ and find an asymptotic formula for the number of its solutions.

Let $\mathcal{P}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this paper, it is proved that, for $12\leqslant b\leqslant 35$ and for every sufficiently large odd integer $N$, the equation…

An integer $n$ is said to be ternary if it is composed of three distinct odd primes. In this paper, we asymptotically count the number of ternary integers $n \leq x$ with the constituent primes satisfying various constraints. We apply our…

For n=1,2,3,... let p_n be the n-th prime. We mainly show that p_n>n+sum_{k=1}^n p_k/k for all n>124, and sum_{k=1}^n kp_k<n^2p_n/3 for all n>30.

Let $X$ be a scheme of finite type over $\mathbf{Z}$. For $p \in \mathcal{P}$ the set of prime numbers, let $N_{X}(p)$ be the number of $\mathbf{F}_{p}$-points of $X/\mathbf{F}_{p}$. For fixed $n\geq 1$ and $a_{1}, \ldots, a_{n} \in…

Let $\mathbb{N}_0$ be a class of natural numbers whose binary expansions contain even numbers of ones. Goldbach's problem in numbers of class $\mathbb{N}_0$ is solved.

Let $\mathcal{P}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this paper, it is proved that for every sufficiently large even integer $N$, the equation \begin{equation*}…

The binary Goldbach conjecture asserts that every even integer greater than $4$ is the sum of two primes. In a preceding paper we have proved that there exists a positive integer $K_\alpha$ such that every even integer $x > p_k^2$ can be…

We formulate some refinements of Goldbach's conjectures based on heuristic arguments and numerical data. For instance, any even number greater than 4 is conjectured to be a sum of two primes with one prime being 3 mod 4. In general, for…

Let $p_1=2, p_2=3, p_3=5, \ldots$ be the consecutive prime numbers, $S_n$ the numerical semigroup generated by the primes not less than $p_n$ and $u_n$ the largest irredundant generator of $S_n$. We will show, that $\bullet$ $u_n\sim3p_n$.…

The idea of generating prime numbers through sequence of sets of co-primes was the starting point of this paper that ends up by proving two conjectures, the existence of infinitely many twin primes and the Goldbach conjecture. The main idea…

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…

In this paper, we compute the size of the exceptional set in a generalized Goldbach problem and show that for a given polynomial $f(x) \in \mathbb{Z}[x]$ with a positive leading coefficient, positive integers $A$, $B$, $g$ and $0 \leq i, j…

For a set $A \subset \mathbb{N}$ we characterize in terms of its density when there exists an infinite set $B \subset \mathbb{N}$ and $t \in \{0,1\}$ such that $B+B \subset A-t$, where $B+B : =\{b_1+b_2\colon b_1,b_2 \in B\}$. Specifically,…

Let P be the set of all primes and P_2=P\cup{p_1p_2: p_1,p_2\in P}$. We prove that the sumset 2^P+P_2={2^p+q: p\in P, q\in P_2} has a positive lower density.

For every even integer N, denote by D_{1,2}(N) the number of representations of N as a sum of a prime and an integer having at most two prime factors. In this paper, we give a new lower bound for D_{1,2}(N).

We estimate the asymptotic density of the set $\bar{A}$ of primes $p$ satisfying the constraint that $p+1$ and $p-1$ have only one prime divisor larger than $3$. We also estimate the density of a maximal subset $\bar{B} \subset \bar{A}$…

Let $g$ be sufficiently large, $b\in\{0,\ldots,g-1\}$, and $\mathcal{S}_b$ be the set of integers with no digit equal to $b$ in their base $g$ expansion. We prove that every sufficiently large odd integer $N$ can be written as $p_1 + p_2 +…

We determine the asymptotic density $\delta_k$ of the set of ordered $k$-tuples $(n_1,...,n_k)\in \N^k, k\ge 2$, such that there exists no prime power $p^a$, $a\ge 1$, appearing in the canonical factorization of each $n_i$, $1\le i\le k$,…

Let $f(n)=\min_{p} |n-p|$, where $p$ is a prime. We show that there is a positive constant $\delta$ such that for any large integer $N$ there exist two positive integers $n_1$ and $n_2$ such that $N=n_1 + n_2$ and $f(n_i)\gg \ln N (\ln\ln…