Related papers: Squarefree numbers in short intervals

In this note, we prove by using T. Estermann's and S. Dimitrov's arguments with an elementary inequality that there are infinitely many $n$ for which all of the numbers $n^2+1,n^2+2$ and $n^2+3$ are squarefree. We also improve the error…

In the present paper we show that there exist infinitely many consecutive square-free numbers of the form $[\alpha n]$, $[\alpha n]+1$, where $\alpha>1$ is irrational number with bounded partial quotient or irrational algebraic number.

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

Every natural number greater than $2$ can be written as the sum of a prime and a square-free number, and recent work has imposed additional divisibility conditions on the square-free number. We overcome limitations in these works to prove…

We give asymptotics for correlation sums linked with the distribution of squarefree numbers in arithmetic progressions over a fixed modulus. As a particular case we improve a result of Blomer concerning the variance.

We prove an asymptotic formula for squarefree in arithmetic progressions with squarefree moduli, improving previous results by Prachar. The main tool is an estimate for counting solutions of a congruence inside a box that goes beyond what…

The main topic of this contribution is the problem of counting square-free numbers not exceeding $n$. Before this work we were able to do it in time (Comparing to the Big-O notation, Soft-O ($\softO$) ignores logarithmic factors)…

It is conjectured that all separable polynomials with integers coefficients, satisfying some local conditions, take infinitely many square free values on integer arguments. But not a single polynomial of degree greater than $3$ is proven to…

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 prove that a positive proportion of squarefree integers are congruent numbers such that the canonical height of the lowest non-torsion rational point on the corresponding elliptic curve satisfies a strong lower bound.

We improve on the best available bounds for the square-free sieve and provide a general framework for its applicability. The failure of the local-to-global principle allows us to obtain results better than those reached by a classical…

Let f be a square-free polynomial in Fq[t][x] where Fq is a field of q elements. We view f as a polynomial in the variable x with coefficients in the ring Fq[t]. We study squarefree values of f in sparse subsets of Fq[t] which are given by…

For a fixed exponent $0 < \theta \leq 1$, it is expected that we have the prime number theorem in short intervals $\sum_{x \leq n < x+x^\theta} \Lambda(n) \sim x^\theta$ as $x \to \infty$. From the recent zero density estimates of Guth and…

Let $M(x)$ be the length of the largest subinterval of $[1,x]$ which does not contain any sums of two squareful numbers. We prove a lower bound \[ M(x)\gg \frac{\ln x}{(\ln\ln x)^2} \] for all $x\geq 3$. The proof relies on properties of…

Let $\mathcal{R}$ be a finite set of integers satisfying appropriate local conditions. We show the existence of long clusters of primes $p$ in bounded length intervals with $p-b$ squarefree for all $b \in \mathcal{R}$. Moreover, we can…

Given a positive integer $n$, we let ${\rm sfp}(n)$ denote the squarefree part of $n$. We determine all positive integers $n$ for which $\max \{ {\rm sfp}(n), {\rm sfp}(n+1), {\rm sfp}(n+2) \} \leq 150$ by relating the problem to finding…

Various authors, including McNew, Nathanson and O'Bryant, have recently studied the maximal asymptotic density of a geometric progression free sequence of positive integers. In this paper we prove the existence of geometric progression free…

Let E be a non-CM elliptic curve defined over Q. For each prime p of good reduction, E reduces to a curve E_p over the finite field F_p. For a given squarefree polynomial f(x,y), we examine the sequences f_p(E) := f(a_p(E), p), whose values…

A word is square-free if it does not contain nonempty factors of the form $XX$. In 1906 Thue proved that there exist arbitrarily long square-free words over a $3$-letter alphabet. It was proved recently [7] that among these words there are…

For any positive integer $n$ along with parameters $\alpha$ and $\nu$, we define and investigate $\alpha$-shifted, $\nu$-offset, floor sequences of length $n$. We find exact and asymptotic formulas for the number of integers in such a…