Related papers: Large strings of consecutive smooth integers

We revisit Schnorr's lattice-based integer factorization algorithm, now with an effective point of view. We present effective versions of Theorem 2 of Schnorr's "Factoring integers and computing discrete logarithms via diophantine…

We give an overview of combinatoric properties of the number of ordered $k$-factorizations $f_k(n,l)$ of an integer, where every factor is greater or equal to $l$. We show that for a large number $k$ of factors, the value of the cumulative…

We determine continuous bijections $f$, acting on a real interval into itself, whose $k$-fold iterate is the quasi-arithmetic mean of all its subsequent iterates from $f^0$ up to $f^n$ (where $0\le k\le n$). Namely, we prove that if at most…

The classical Erd\H{o}s-P\'{o}sa theorem states that for each positive integer k there is an f(k) such that, in each graph G which does not have k+1 disjoint cycles, there is a blocker of size at most f(k); that is, a set B of at most f(k)…

When $k$ is a constant at least $3$, a sequence $S$ of positive integers is called $k$-GP-free if it contains no nontrivial $k$-term geometric progressions. Beiglb\"ok, Bergelson, Hindman and Strauss first studied the existence of a $…

In the integer case, the Smarandache function of a positive integer $n$ is defined to be the smallest positive integer $k$ such that $n$ divides the factorial $k!$. In this paper, we first define a natural order for polynomials in…

Using a sieve-theoretic argument, we show that almost all gaps $(p_n, p_{n+1})$ between consecutive primes $p_n, p_{n+1}$ contain a natural number $m$ whose least prime factor $p(m)$ is at least the length $p_{n+1} - p_n$ of the gap,…

One of the most classical results in Ramsey theory is the theorem of Erd\H{o}s and Szekeres from 1935, which says that every sequence of more than $k^2$ numbers contains a monotone subsequence of length $k+1$. We address the following…

In some applications, one is interested in reconstructing a function $f$ from its Fourier series coefficients. The problem is that the Fourier series is slowly convergent if the function is non-periodic, or is non-smooth. In this paper, we…

Sharpening (a particular case of) a result of Szemeredi and Vu and extending earlier results of Sarkozy and ourselves, we find, subject to some technical restrictions, a sharp threshold for the number of integer sets needed for their sumset…

Let $f_1,\dots,f_k\in\mathbb{R}[X]$ be polynomials of degree at most $d$ with $f_1(0)=\dots=f_k(0)=0$. We show that there is an integer $n<x$ such that the fractional parts $\|f_i(n)\|\ll x^{c/k}$ for all $1\le i\le k$ and for some constant…

Let \(d_k(p)\) denote the natural density of positive integers whose \(k\)-th smallest prime divisor is \(p\). Erd\H{o}s asked whether, for each fixed \(k\), the sequence \(p\mapsto d_k(p)\) is unimodal as \(p\) ranges over the primes.…

We prove an explicit analogue of Legendre's conjecture for almost primes. Namely, for every integer $n \geq 1$, the interval $(n^2,(n+1)^2)$ contains an integer having at most $3$ prime factors, counted with multiplicity. This improves the…

Solving a problem by Erd\H{o}s, we prove that every positive integer $n$ can be written as a sum $$n = b_{1} + b_{2} + \ldots + b_{r}$$ of distinct $3$-smooth integers with $1 \le b_{1} < b_{2} < \ldots < b_{r} < 6b_{1}$.

A family of sets is said to be \emph{intersecting} if any two sets in the family have nonempty intersection. In 1973, Erd\H{o}s raised the problem of determining the maximum possible size of a union of $r$ different intersecting families of…

An n-ary k-radius sequence is a finite sequence of elements taken from an alphabet of size n such that any two distinct elements of the alphabet occur within distance k of each other somewhere in the sequence. These sequences were…

We give some theoretical and computational results on "random" harmonic sums with prime numbers, and more generally, for integers with a fixed number of prime factors.

For each integer k >= 2, let F(k) denote the largest n for which there exists a permutation \sigma \in S_n, all of whose patterns of length k are distinct. We prove that F(k) = k + \lfloor \sqrt{2k-3} \rfloor + e_k, where e_k \in {-1,0} for…

For any positive integer $k$, we show that infinitely often, perfect $k$-th powers appear inside very long gaps between consecutive prime numbers, that is, gaps of size $$ c_k \frac{\log p \log_2 p \log_4 p}{(\log_3 p)^2}, $$ where $p$ is…

We study the compositions of an integer n whose part sizes do not exceed a fixed integer k. We use the methods of analytic combinatorics to obtain precise asymptotic formulas for the number of such compositions, the total number of parts…