Related papers: On the least almost-prime in arithmetic progressio…

Assuming a uniform $q$-variant of the prime $k$-tuple conjecture, we compute moments of the number of primes in arithmetic progressions to a large modulus $q$ as the residue classes vary. Consequently, depending on the size of $\varphi(q)$,…

In this paper, the estimation formula of the number of primes in a given interval is obtained by using the prime distribution property. For any prime pairs $p>5$ and $ q>5 $, construct a disjoint infinite set sequence $A_1, A_2, \ldots,…

Given $d\geq 2$, we show that the number of approximates $\frac{1}{q}\mathbf{p}\in \mathbb{Q}^d$ of $\mathbf{x}\in\mathbb{R}^d$ satisfying $|q\mathbf{x}-\mathbf{p}|\leq cq^{-\frac{1}{d}}$ with denominator $1\leq q < T$ decays to the…

We obtain an asymptotic formula for the average value of the divisor function over the integers $n \le x$ in an arithmetic progression $n \equiv a \pmod q$, where $q=p^k$ for a prime $p\ge 3$ and a sufficiently large integer $k$. In…

Let $\sigma(n)$ be the sum of the positive divisors of $n$. A positive integer $n$ is said to be $2$-near perfect when $\sigma(n)=2n+d_1+d_2$, where $d_1$ and $d_2$ are distinct positive divisors of $n$. We show that there are no odd…

Let $k \geq 3$ and let $L_i(n) = A_in + B_i$ be some linear forms such that $A_i$ and $B_i$ are integers. Define ${\mathcal{P}(n) = \prod_{i=1}^k L_i(n)}$. For each $k$ it is known that $\Omega (\mathcal{P} (n) ) \leq \rho_k$ infinitely…

In RSA cryptography numbers of the form $pq$, with $p$ and $q$ two distinct proportional primes play an important role. For a fixed real number $r>1$ we formalize this by saying that an integer $pq$ is an RSA-integer if $p$ and $q$ are…

Given an integer $m \geq 2$ and a sufficiently large $q$, we apply a variant of the Maynard--Tao sieve weight to establish the existence of an arithmetic progression with common difference $q$ for which the $m$-th least prime in such…

Famous Zaremba's conjecture (1971) states that for each positive integer $q\geq2$, there exists positive integer $1\leq a <q$, coprime to $q$, such that if you expand a fraction $a/q$ into a continued fraction $a/q=[a_1,\ldots,a_n]$, all of…

For any positive integer r, let pi_{2r}(x) denote the number of prime pairs (p, p+2r) with p not exceeding (large) x. According to the prime-pair conjecture of Hardy and Littlewood, pi_{2r}(x) should be asymptotic to 2C_{2r}li_2(x) with an…

Let $E(X)$ denote the number of even integers below $X$ which are not a sum of two primes. We prove the bound $E(X)=O(X^{\frac{7}{10}})$, where the implicit constant is ineffective. The method applied here also leads to $P(q)=O(q^5)$, where…

In the present paper we prove that there exist infinitely many arithmetic progressions of three different primes $p_1,p_2,p_3=2p_2-p_1$ such that $p_1=x_1^2 + y_1^2 +1$, $p_2=x_2^2 + y_2^2 +1$.

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…

This note investigates the prime values of the polynomial $f(t)=qt^2+a$ for any fixed pair of relatively prime integers $ a\geq 1$ and $ q\geq 1$ of opposite parity. For a large number $x\geq1$, an asymptotic result of the form $\sum_{n\leq…

Fix coprime natural numbers $a,q$. Assuming the Prime $k$-tuple Conjecture, we show that there exist arbitrarily long arithmetic progressions of Carmichael numbers, each of which lies in the reduced residue class $a$ mod $q$ and is a…

Denote by $\pi(x;q,a)$ the number of primes $p\leqslant x$ with $p\equiv a\bmod q.$ We prove new upper bounds for $\pi(x;q,a)$ when $q$ is a large prime very close to $\sqrt{x}$, improving upon the classical work of Iwaniec (1982). The…

For any real number $t$, let $[t]$ denote the integer part of $t$. In this paper it is proved that if $1<c<\frac{247}{238}$, then for sufficiently large integer $N$, the equation \[\left[p^{c}\right]+\left[m^{c}\right]=N\] has a solution in…

We prove that there are infinitely many solutions of $$ |\lambda_0+\lambda_1p+\lambda_2P_r|<p^{-\tau}, $$ where $r=3,$ $\tau=\frac1{118}$, and $\lambda_0$ is an arbitrary real number and $\lambda_1,\lambda_2\in\BR$ with $\lambda_2\neq0$ and…

We prove lower bounds for the number of primes $p \leq N + b$ such that $p-b$ is divisible by $2^{k(N)}$ and has at most $k$ odd prime factors ($k \geq 2$), assuming $2^{k(N)} \leq N^\theta$ for some $\theta > 0$ depending on $k$. The proof…

Let $p,q>1$ be two relatively prime integers and $\mathbb{N}$ the set of nonnegative integers. Let $f_{p,q}(n)$ be the number of different expressions of $n$ written as a sum of distinct terms taken from…