Let ${𝒫}_2$ denote a positive integer with at most $2$ prime factors, counted according to multiplicity. For integers $a$, $q$ such that $(a,q)=1$, let ${𝒫}_2(q,a)$ denote the least ${𝒫}_2$ in the arithmetic progression ${\{nq+a\}}_{n=1}^{\infty }$. It is proved that for sufficiently large $q$, we have $${𝒫}_2(q,a)\ll q^{1.825}.$$ This result constitutes an improvement upon that of J. Li, M. Zhang and Y. Cai (2023), who obtained ${𝒫}_2(q,a)\ll q^{1.8345}.$

We prove that almost all positive even integers $n$ can be represented as $p_2^2+p_3^3+p_4^4+p_5^5$ with $|p_k^k-\frac{1}{4}N|\le N^{1-1/54+\epsilon }$ for $2\le k\le 5$. As a consequence, we show that each sufficiently large odd integer $N$ can be written as $p_1+p_2^2+p_3^3+p_4^4+p_5^5$ with $|p_k^k-\frac{1}{5}N|\le N^{1-1/54+\epsilon }$ for $1\le k\le 5$.

Let $a$ and $b\in \mathbb{N}$. Denote by $R_{a,b}$ the set of all integers $n>1$ whose canonical prime representation $n=p_1^{{\alpha }_1}{p}_2^{{\alpha }_2}\cdots p_r^{{\alpha }_r}$ has all exponents ${\alpha }_i$ $(1\le i\le r)$ being a multiple of $a$ or belonging to the arithmetic progression $at+b$, $t\in {\mathbb{N}}_0:=\mathbb{N}\cup \left\{0\right\}$. All integers in $R_{a,b}$ are called generalized square-full integers. Using the exponent pair method, an upper bound for character sums over generalized square-full integers is derived. An application on the distribution of generalized square-full integers in an arithmetic progression is given. ...

Let ${𝒫}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. Suppose that $a$ and $q$ are positive integers satisfying $(a,q)=1$. Denote by ${𝒫}_2(a,q)$ the least almost-prime ${𝒫}_2$ which satisfies ${𝒫}_2\equiv a(modq)$. It is proved that for sufficiently large $q$, there holds $${𝒫}_2(a,q)\ll q^{1.8345}.$$ This result constitutes an improvement upon that of Iwaniec (1982), who obtained the same conclusion, but for the range $1.845$ in place of $1.8345$.

Given an integer $n\ge 3$, let $u_1,...,u_n$ be pairwise coprime integers $\ge 2$, $𝒟$ a family of nonempty proper subsets of $\{1,...,n\}$ with “enough” elements, and $\epsilon$ a function $𝒟\to \{\pm 1\}$. Does there exist at least one prime $q$ such that $q$ divides ${\prod }_{i\in I}{u}_i-\epsilon \left(I\right)$ for some $I\in 𝒟$, but it does not divide $u_1\cdots u_n$? We answer this question in the positive when the $u_i$ are prime powers and $\epsilon$ and $𝒟$ are subjected to certain restrictions. We use the result to prove that, if ${\epsilon }_0\in \{\pm 1\}$ and $A$ is a set of three or more primes that contains all prime divisors of any...

Consider a recurrence sequence ${\left(x_k\right)}_{k\in \mathbb{Z}}$ of integers satisfying $x_{k+n}=a_{n-1}{x}_{k+n-1}+...+a₁x_{k+1}+a₀x_k$, where $a₀,a₁,...,a_{n-1}\in \mathbb{Z}$ are fixed and a₀ ∈ -1,1. Assume that $x_k>0$ for all sufficiently large k. If there exists k₀∈ ℤ such that $x_{k₀}<0$ then for each negative integer -D there exist infinitely many rational primes q such that $q|x_k$ for some k ∈ ℕ and (-D/q) = -1.

Let $𝒱$ be a fixed algebraic variety defined by $m$ polynomials in $n$ variables with integer coefficients. We show that there exists a constant $C\left(𝒱\right)$ such that for almost all primes $p$ for all but at most $C\left(𝒱\right)$ points on the reduction of $𝒱$ modulo $p$ at least one of the components has a large multiplicative order. This generalises several previous results and is a step towards a conjecture of B. Poonen.

Let $\mathbb{Z}$ be the set of integers, ${\mathbb{N}}_0$ the set of nonnegative integers and $F(x_1,x_2)=u_1{x}_1+u_2{x}_2$ be a binary linear form whose coefficients $u_1$, $u_2$ are nonzero, relatively prime integers such that $u_1{u}_2\ne \pm 1$ and $u_1{u}_2\ne -2$. Let $f:\mathbb{Z}\to {\mathbb{N}}_0\cup \left\{\infty \right\}$ be any function such that the set $f^{-1}\left(0\right)$ has asymptotic density zero. In 2007, M. B. Nathanson (2007) proved that there exists a set $A$ of integers such that $r_{A,F}\left(n\right)=f\left(n\right)$ for all integers $n$, where $r_{A,F}\left(n\right)=\left|\{(a,a^{\text{'}}):n=u_{1}a+u_2{a}^{\text{'}}:a,a^{\text{'}}\in A\}\right|$. We add the structure of difference for the binary linear form $F(x_1,x_2)$.