Related papers: An explicit version of Chen's theorem and the line…

We prove new results on the additive theory of reversed primes $\overleftarrow{p}$; that is, primes $p$ which are written backwards in a fixed base $b\geq 2$. In particular, we study a variant of Goldbach's conjecture, looking at…

We say the sets of nonnegative integers A and B are additive complements if their sum contains all sufficiently large integers. In this paper we prove a conjecture of Chen and Fang about additive complement of a finite set.

In this paper, I proved that $$N=p_1+p_2+2p_3, p_1\sim N/2, p_2\sim N/2, p_3=o(N),$$ where $N$ is a large even number, and $p_i\ (i=1,2,3)$ are odd primes.

For $n \geq 1$ let $ p_n $ denote the $n^{\rm th}$ prime number. Let $$S= \{1,7,11,13,17,19,23,29 \},$$ the set of positive integers which are both less than and relatively prime to $30.$ For $ x \geq 0,$ let \\ $T_x := \{ 30x+i \; | \; i…

The aim of this note is to show that any even perfect number, other than $6$, can be written as the sum of 5 cubes of natural numbers. We also conjecture that any even perfect number, other than $6$, can be written as the sum of only 3…

In this short paper we present an elementary proof of the infinitude of primes. Our proof is similar in spirit to Euler's proof that the reciprocals of primes diverges and only uses tools from elementary number theory and calculus. In…

Let $\mathcal{P}$ be the set of all primes and $\pi(x)$ be the number of primes up to $x$. For any $n\ge 2$, let $P^+(n)$ be the largest prime factor of $n$. For $0<c<1$, let $$T_c(x)=\#\{p\le x:p\in \mathcal{P},P^+(p-1)\ge p^c\}.$$ In this…

In this paper we continue our study, begun in part I, of the exceptional set of integers, not restricted by elementary congruence conditions, which cannot be represented as sums of three or four squares of primes. We correct a serious…

We show that all natural numbers $n\equiv 4\pmod 6$ are the sum of two Chen primes (primes $p$ such that $p+2$ has at most two prime factors), apart from a power-saving set of exceptions. This improves on various previous results and is…

Let $\delta > 1/2$. We prove that if $A$ is a subset of the primes such that the relative density of $A$ in every reduced residue class is at least $\delta$, then almost all even integers can be written as the sum of two primes in $A$. The…

Is is shown that all but O(x^{23207/23240}) even integers N<x can be written as the sum of a square, a cube, a forth and a fifth power of a prime.

Improving earlier estimates of several authors we show that the number E(X) of Goldbach exceptional even integers (that is, even integers which cannot be written as the sum of two primesw) below X satisfies tho bound E(X) < X^0.72 for…

In the present paper we prove that under the assumption of the GRH (Generalized Riemann Hypothesis) each sufficiently large odd integer can be expressed as the sum of a prime and two isolated primes.

The famous strongly binary Goldbach's conjecture asserts that every even number $2n \geq 8$ can always be expressible as the sum of two distinct odd prime numbers. We use a new approach to dealing with this conjecture. Specifically, we…

We establish that, for almost all natural numbers $N$, there is a sum of two positive integral cubes lying in the interval $[N-N^{7/18+\epsilon},N]$. Here, the exponent $7/18$ lies half way between the trivial exponent $4/9$ stemming from…

Generalized octagonal numbers are those $p_8(x)=x(3x-2)$ with $x\in\mathbb Z$. In this paper we mainly show that every positive integer can be written as the sum of four generalized octagonal numbers one of which is odd. This result is…

A natural number $n$ is called semi-prime if it is a product of two primes or a square of a prime. We denote $\mathbb{P}_2$ the set of all semi-primes. Our goal is to prove that for fixed integer number $a$ and sufficiently large $x$ the…

We introduce a simple sieve-theoretic approach to studying partial sums of multiplicative functions which are close to their mean value. This enables us to obtain various new results as well as strengthen existing results with new proofs.…

We prove that every positive integer $n$ which is not equal to $1$, $2$, $3$, $6$, $11$, $30$, $155$, or $247$ can be represented as a sum of a squarefree number and a prime not exceeding $\sqrt{n}$.

In this paper we give an additive representation of the factorial, which can be proven by a simple quick analytical argument. We also present some generalizations, which are linked, on the one hand to an arithmetical theorem proven by Euler…