Related papers: Smooth squarefree and square-full integers in arit…

We show that for any $\varepsilon > 0$ and a sufficiently large cube-free $q$, any reduced residue class modulo $q$ can be represented as a product of $14$ integers from the interval $[1, q^{1/4e^{1/2} + \varepsilon}]$. The length of the…

We study representation of square-free polynomials in the polynomial ring F[t] over a finite field F by polynomials in F[t][x]. This is a function field version of the well-studied problem of representing squarefree integers by integer…

Let $p>2$ be prime and $g$ a primitive root modulo $p$. We present an argument for the fact that discrete logarithms of the numbers in any arithmetic progression are uniformly distributed in $[1,p]$ and raise some questions on the subject.

This note presents new results for the squarefree value sets of quartic polynomials over the integers.

Every integer greater than two can be expressed as the sum of a prime and a square-free number. Expanding on recent work, we provide explicit and asymptotic results when divisibility conditions are imposed on the square-free number. For…

We prove new results related to the digital reverse $\overleftarrow{n}$ of a positive integer $n$ in a fixed base $b$. First we show that for $b\geq 26000$, there exists infinitely many primes $p$ such that $\overleftarrow{p}$ is…

The number of square-free integers in $x$ consecutive values of any polynomial $f$ is conjectured to be $c_fx$, where the constant $c_f$ depends only on the polynomial $f$. This has been proven for degrees less or equal to 3. Granville was…

Fix $a \in \mathbb{Z}$, $a\notin \{0,\pm 1\}$. A simple argument shows that for each $\epsilon > 0$, and almost all (asymptotically 100% of) primes $p$, the multiplicative order of $a$ modulo $p$ exceeds $p^{\frac12-\epsilon}$. It is an…

We establish estimates for the number of ways to represent any reduced residue class as a product of a prime and an integer free of small prime factors. Our best results is conditional on the Generalised Riemann hypothesis (GRH). As a…

In the context of global optimization and mixed-integer non-linear programming, generalizing a technique of D'Ambrosio, Fampa, Lee and Vigerske for handling the square-root function, we develop a virtuous smoothing method, using cubics,…

Conceptually, a rough number is a positive integer with no small prime factors. Formally, for real numbers $x$ and $y$, let $\Phi(x,y)$ denote the number of positive integers at most $x$ with no prime factors less than $y$. In this paper we…

We prove a modification as well as an improvement of a result of K. Ford, D. R. Heath-Brown and S. Konyagin concerning prime avoidance of square-free numbers and perfect powers of prime numbers.

We provide examples of multiplicative functions $f$ supported on the $k$-free integers such that at primes $f(p)=\pm 1$ and such that the partial sums of $f$ up to $x$ are $o(x^{1/k})$. Further, if we assume the Generalized Riemann…

We develop a sieve that can detect primes in multiplicatively structured sets under certain conditions. We apply it to obtain a new $L$-function free proof of Linnik's problem of bounding the least prime $p$ such that $p\equiv a\pmod q$…

We obtain new bounds of exponential sums modulo a prime $p$ with sparse polynomials $a_0x^{n_0} + \cdots + a_{\nu}x^{n_\nu}$. The bounds depend on various greatest common divisors of exponents $n_0, \ldots, n_\nu$ and their differences. In…

Let $p$ be an odd prime. In this paper we investigate quadratic residues modulo $p$ and related permutations, congruences and identities. If $a_1<\ldots<a_{(p-1)/2}$ are all the quadratic residues modulo $p$ among $1,\ldots,p-1$, then the…

Let $u\ne \pm 1$, and $v\ne \pm 1$ be a pair of fixed relatively prime squarefree integers, and let $d\geq 1$, and $e \geq1$ be a pair of fixed integers. It is shown that there are infinitely many primes $p\geq 2$ such that $u$ and $v$ have…

Let $p$ be a large odd prime, let $x=\log p)(\log\log p)^{3+\varepsilon}$ and let $q\ll\log\log p$ be an integer, where $\varepsilon>0$ is a small number. This note proves the existence of small prime quadratic residues and small prime…

We primarily investigate congruences modulo $p$ for finite sums of the form $\sum_k\binom{rk}{k}x^k/k$ over the ranges $0<k<p$ and $0<k<p/r$, where $p$ is a prime larger than the positive integer $r$. Here $x$ is an indeterminate, thus…

We obtain the analog of the Bombieri-Vinogradov theorem for square moduli up to any power of x less than 1/2.