Related papers: A note on primes in certain residue classes

Let $p$ be a prime number. We say that a positive integer $n$ is a Sylow $p$-number if there exists a finite group having exactly $n$ Sylow $p$-subgroups. When $p=2$, every odd integer is a Sylow $2$-number. In contrast, when $p$ is odd,…

We consider the Diophantine equation $$ a!b! = c! $$ due to Erd\H{o}s, where we assume $a \leq b$. It is widely believed that there are only finitely many nontrivial solutions, and considerable work has been dedicated to showing this. In…

Given an integer $n \ge 3$, let $u_1, \ldots, u_n$ be pairwise coprime integers $\ge 2$, $\mathcal D$ a family of nonempty proper subsets of $\{1, \ldots, n\}$ with "enough" elements, and $\varepsilon$ a function $ \mathcal D \to \{\pm…

Let $ x\geq 1 $ be a large number, let $ [x]=x-\{x\} $ be the largest integer function, and let $ \varphi(n)$ be the Euler totient function. The asymptotic formula for the new finite sum over the primes $ \sum_{p\leq…

We study an asymptotic formula for average orders of Goldbach representations of an integer as the sum of k primes. We extend the existing result for k=2 to a general k, for which we obtain a better error term. Moreover, we prove an…

In this paper, we characterize the odd positive integers $n$ satisfying the congruence $\sum_{j=1} ^ {n-1} j^{\frac{n-1}{2}}\equiv 0 \pmod n$. We show that the set of such positive integers has an asymptotic density which turns out to be…

Let $P$ be a subset of the primes of lower density strictly larger than $\frac12$. Then, every sufficiently large even integer is a sum of four primes from the set $P$. We establish similar results for $k$-summands, with $k\geq 4$, and for…

Let $s\geq 8$ be an integer and $P$ be a set of primes with relative lower density greater than $\sqrt{1-\min\{s,16\}/32}$. We prove that every sufficiently large integer $n\equiv s({\rm mod}24)$ can be represented by a sum of $s$ squares…

Given a finite family F of linear forms with integer coefficients, and a compact abelian group G, an F-free set in G is a measurable set which does not contain solutions to any equation L(x)=0 for L in F. We denote by d_F(G) the supremum of…

For a prime number $p$ and a sequence of integers $a_0,\dots,a_k\in \{0,1,\dots,p\}$, let $s(a_0,\dots,a_k)$ be the minimum number of $(k+1)$-tuples $(x_0,\dots,x_k)\in A_0\times\dots\times A_k$ with $x_0=x_1+\dots + x_k$, over subsets…

Let $A(p,n,k)$ be the number of $p$-tuples of commuting permutations of $n$ elements whose permutation action results in exactly $k$ orbits or connected components. We formulate the conjecture that, for every fixed $p$ and $n$, the…

Let $pod_{\ell}(n)$ be the number of $\ell$-regular partitions of $n$ with distinct odd parts. In this article, prove that for any positive integer $k$, the set of non-negative integers $n$ for which $pod_{\ell}(n)\equiv 0 \pmod{p^{k}}$ has…

Motivated by a question of V. Bergelson and F. K. Richter (2017), we obtain asymptotic formulas for the number of relatively prime tuples composed of positive integers $n\le N$ and integer parts of polynomials evaluated at $n$. The error…

For K a cyclic cubic number field with odd class number containing a unit w such that Norm(w)=Norm(1-w)=-1, we prove that the density of rational primes p that satisfy the given spin relation is equal to 1/2. Furthermore, we prove that this…

In this paper we show that if $A$ is a subset of the primes with positive relative density $\delta$, then $A+A$ must have positive upper density $C_1\delta e^{-C_2(\log(1/\delta))^{2/3}(\log\log(1/\delta))^{1/3}}$ in $\mathbb{N}$. Our…

Let $\pi(x;\gamma_1,\gamma_2)$ denote the number of primes $p$ with $p\leqslant x$ and $p=\lfloor n^{1/\gamma_1}_1\rfloor=\lfloor n^{1/\gamma_2}_2\rfloor$, where $\lfloor t\rfloor$ denotes the integer part of $t\in\mathbb{R}$ and…

Let $h,k \ge 2$ be integers. We say a set $A$ of positive integers is an asymptotic basis of order $k$ if every large enough positive integer can be represented as the sum of $k$ terms from $A$. A set of positive integers $A$ is called…

Let k>1 be an integer and let p be a prime. We show that if $p^a\le k<2p^a$ or $k=p^aq+1$ (with 2q<p) for some a=1,2,..., then the set {\binom{n}{k}: n=0,1,2,...} is dense in the ring Z_p of p-adic integers, i.e., it contains a complete…

For a subset $\mathcal A\subset \mathbb N$, let $p_{\mathcal A}(n)$ denote the restricted partition function which counts partitions of $n$ with all parts lying in $\mathcal A$. In this paper, we use a variation of the Hardy-Littlewood…

Let A be a nonempty finite set of relatively prime positive integers, and let p_A(n) denote the number of partitions of n with parts in A. An elementary arithmetic argument is used to obtain an asymptotic formula for p_A(n).