Related papers: Waring-Goldbach problem in short intervals

We establish two new Waring--Goldbach type representations: every sufficiently large odd integer $n$ can be expressed as \[ n = p_1^2 + p_2^2 + p_3^3 + p_4^3 + p_5^5 + p_6^6 + p_7^c, \] where each $p_i$ is prime and $c \in \{6,7\}$.

Let $\mathcal{P}$ denote the set of all primes. $P_{1},P_{2},P_{3}$ are three subsets of $\mathcal{P}$. Let $\underline{\delta}(P_{i})$ $(i=1,2,3)$ denote the lower density of $P_{i}$ in $\mathcal{P}$, respectively. It is proved that if…

We show that once $\theta>17/30$, every sufficiently long interval $[x,x+x^\theta]$ contains many $k$-term arithmetic progressions of primes, uniformly in the starting point $x$. More precisely, for each fixed $k\ge3$ and $\theta>17/30$,…

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…

Let $k\ge 2$ be a positive integer and $P^+(n)$ the greatest prime factor of a positive integer $n$ with convention $P^+(1)=1$. For any $\theta\in \left[\frac 1{2k},\frac{17}{32k}\right)$, set…

Baker, Harman, and Pintz showed that a weak form of the Prime Number Theorem holds in intervals of the form $[x-x^{0.525},x]$ for large $x$. In this paper, we extend a result of Maynard and Tao concerning small gaps between primes to…

In the present paper we prove that every sufficiently large odd integer $N$ can be represented in the form \begin{equation*} N=p_1+p_2+p_3\,, \end{equation*} where $p_1,p_2,p_3$ are primes, such that $p_1=x^2 + y^2 +1$, $p_2=[n^c]$.

Legendre's conjecture states that there is a prime number between n^2 and (n+1)^2 for every positive integer n. We consider the following question : for all integer n>1 and a fixed integer k<=n does there exist a prime number such that kn <…

Let $k \geq 2$ and $b \geq 3$ be integers, and suppose that $d_1, d_2 \in \{0,1,\dots, b - 1\}$ are distinct and coprime. Let $\mathcal{S}$ be the set of non-negative integers, all of whose digits in base $b$ are either $d_1$ or $d_2$. Then…

Let $q$ be a prime power, and let $r=nk+1$ be a prime such that $r\nmid q$, where $n$ and $k$ are positive integers. Under a simple condition on $q$, $r$ and $k$, a Gauss period of type $(n,k)$ is a normal element of $\Bbb F_{q^n}$ over…

Let $n$ be a positive integer greater than $2$. We define \textit{the Proth numerical semigroup}, $P_{k}(n)$, generated by $\{k 2^{n+i}+1 \,\mid\, i \in \mathbb{N}\}$, where $k$ is an odd positive number and $k < 2^{n}$. In this paper, we…

We give an upper bound for the minimum $s$ with the property that every sufficiently large integer can be represented as the sum of $s$ positive $k$-th powers of integers represented as the sum of three positive cubes for the cases $2\leq…

Let $G(k)$ denote the least number $s$ such that every sufficiently large natural number is the sum of at most $s$ positive integral $k$th powers. We show that $G(7)\le 31$, $G(8)\le 39$, $G(9)\le 47$, $G(10)\le 55$, $G(11)\le 63$,…

We show that whenever $s>k(k+1)$, then for any complex sequence $(\mathfrak a_n)_{n\in \mathbb Z}$, one has $$\int_{[0,1)^k}\left| \sum_{|n|\le N}\mathfrak a_ne(\alpha_1n+\ldots +\alpha_kn^k) \right|^{2s}\,{\rm d}{\mathbf \alpha}\ll…

We study the representations of large integers $n$ as sums $p_1^2 + ... + p_s^2$, where $p_1,..., p_s$ are primes with $| p_i - (n/s)^{1/2} | \le n^{\theta/2}$, for some fixed $\theta < 1$. When $s = 5$ we use a sieve method to show that…

Let $1<c<d$ be two relatively prime integers, $g_{c,d}=cd-c-d$ and $\mathbb{P}$ is the set of primes. For any given integer $k \geq 1$, we prove that $$\#\left\{p^k\le g_{c,d}:p\in \mathbb{P}, ~p^k=cx+dy,~x,y\in \mathbb{Z}_{\geqslant0}…

Assuming that the Generalized Riemann Hypothesis (GRH) holds, we prove an explicit formula for the number of representations of an integer as a sum of $k\geq 5$ primes. Our error terms in such a formula improve by some logarithmic factors…

Let $k\in \mathbb{N}$ and $s\geq k(\log k+3.20032)$. Let $\mathbb{N}_{0}^{k}$ be the set of $k$-th powers of nonnegative integers. Assume that $\psi$ is an increasing function tending to infinity with $\psi(x)=o(\log x)$ and satifying some…

Let $k\ge 1$ be an integer. A positive integer $n$ is $k$-\textit{gleeful} if $n$ can be represented as the sum of $k$th powers of consecutive primes. For example, $35=2^3+3^3$ is a $3$-gleeful number, and $195=5^2+7^2+11^2$ is $2$-gleeful.…

Let $\mathcal{P}=\{p_1,p_2,...\}$ be the set of all odd primes arranged in increasing order. A Goldbach partition of the even integer $2k>4$ is a way of writing it as a sum of two primes from $\mathcal{P}$ without regard to order. Let…