Related papers: A density version of Waring-Goldbach problem

Let $\mu$ be a given Borel measure on $\K\subseteq\R^n$ and let $y=(y_\alpha)$, $\alpha\in\N^n$, be a given sequence. We provide several conditions linking $y$ and the moment sequence $z=(z_\alpha)$ of $\mu$, for $y$ to be the moment…

In the polynomial ring $T=k[y_1,...,y_n]$, with $n>1$, we bound the multiplicity of homogeneous radical ideals $I\subset (y_1^{a_1},...,y_n^{a_n})$ such that $T/I$ is a graded $k$-algebra with Krull dimension one. As a consequence we solve…

For each positive integer $n$, let $g_\Delta(n)$ be the smallest positive integer $g$ such that every complete quadratic polynomial in $n$ variables which can be represented by a sum of odd squares is represented by a sum of at most $g$ odd…

Let a,f and g be integers, with a and f coprime. Under the generalized Riemann hypothesis it follows from work of Hooley and Lenstra that the set of primes p such that p=a(mod f) and g is primitive root mod p has a natural density. In this…

In this paper we present some observations about the well-known Goldbach conjecture. In particular we list and interpret some numerical results which allow us to formulate a relation between prime numbers and even integers. We can also…

We find an upper bound for the sum $\sum_{x<n\leq 2x}\textbf{1}_{\mathbb{P}}(n+h_{i_{1}})\cdots\textbf{1}_{\mathbb{P}}(n+h_{i_{m+1}})w_{n}$, where $(h_{i_{1}},...,h_{i_{m+1}})$ is any $(m+1)$-tuple of elements in the admissible set…

For integer $k \geq 1$, let $S_k(n)$ denote the sum of the $k$th powers of the first $n$ positive integers. In this paper, we derive a new formula expressing $2^{2k}$ times $S_{2k}(n)$ as a sum of $k$ terms involving the numbers in the…

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*}…

Let $K$ be a number field and $G$ a finitely generated torsion-free subgroup of $K^\times$. Given a prime $\mathfrak p$ of $K$ we denote by ${\rm ind}_{\mathfrak p}(G)$ the index of the subgroup $(G\bmod\mathfrak p)$ of the multiplicative…

We consider the exceptional set in the binary Goldbach problem for sums of two almost twin primes. Our main result is a power-saving bound for the exceptional set in the problem of representing $m=p_1+p_2$ where $p_1+2$ has at most $2$…

A permutation of n letters is k-prolific if each (n-k)-subset of the letters in its one-line notation forms a unique pattern. We present a complete characterization of k-prolific permutations for each k, proving that k-prolific permutations…

Let G be the product of an abelian variety and a torus defined over a number field K. Let R_1,..., R_n be points in G(K). Let l be a rational prime and let a_1,..., a_n be non-negative integers. Consider the set of primes p of K satisfying…

The Waring problem of forms concerns the expression of homogeneous multivariate polynomials as sums of powers of linear forms. This paper focuses on complex binary forms, and we solve the Waring problem for them using basic tools in algebra…

The set of integers which can be written as the sum of four prime cubes has lower density at least $0.009664$. This improves earlier bounds of $0.003125$ by Ren and $0.005776$ by Liu.

We consider three extremal problems about the number of copies of a fixed graph in another larger graph. First, we correct an error in a result of Reiher and Wagner and prove that the number of $k$-edge stars in a graph with density $x \in…

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 compute all primes up to $6.25\times 10^{28}$ of the form $m^2+1$. Calculations using this list verify, up to our bound, a less famous conjecture of Goldbach. We introduce `Goldbach champions' as part of the verification process and…

Let $q$ be a prime. We give an elementary proof of the fact that for any $k\in\mathbb{N}$, the proportion of $k$-element subsets of $\mathbb{Z}$ that contain a $q^{th}$ power modulo almost every prime, is zero. This result holds regardless…

In this paper, we build some ergodic theorems involving function $\Omega$, where $\Omega(n)$ denotes the number of prime factors of a natural number $n$ counted with multiplicities. As a combinatorial application, it is shown that for any…

We prove that there exists a k_0>0 such that every sufficiently large odd integer n with 3\mid n can be represented as p_1+p_2+p_3, where p_1,p_2 are Chen's primes and p_3 is a prime with p_3+2 has at most k_0 prime factors.