Related papers: Large strings of consecutive smooth integers

In the present work we prove a common generalization of Maynard-Tao's recent result about consecutive bounded gaps between primes and on the Erd\H{o}s-Rankin bound about large gaps between consecutive primes. The work answers in a strong…

The set of short intervals between consecutive primes squared has the pleasant---but seemingly unexploited---property that each interval $s_k:=\{p_k^2, \dots,p_{k+1}^2-1\}$ is fully sieved by the $k$ first primes. Here we take advantage of…

We present and analyze an algorithm to enumerate all integers $n\le x$ that can be written as the sum of consecutive $k$th powers of primes, for $k>1$. We show that the number of such integers $n$ is asymptotically bounded by a constant…

Recent work has established that, for every positive integer $k$, every $n$-node graph has a $(2k-1)$-spanner on $O(f^{1-1/k} n^{1+1/k})$ edges that is resilient to $f$ edge or vertex faults. For vertex faults, this bound is tight. However,…

Let $p_n$ denote the $n$th prime and $g_n:=p_{n+1}-p_n$ the $n$th prime gap. We demonstrate the existence of infinitely many values of $n$ for which $g_n>g_{n+1}>\cdots>g_{n+m}$ with $m\gg \log\log\log n$ and similarly for the reversed…

We establish the order of the maximum length of an increasing sequence, bounded by $n$, in which the largest prime divisor of the elements form a decreasing sequence.

Let $f(m)$ be the largest integer such that for every set $A = \{a_1 < \cdots < a_m\}$ of $m$ positive integers and every open interval $I$ of length $2a_m$, there exist at least $f(m)$ disjoint pairs $(a, b)$ with $a \in A$ dividing $b \in…

Let P be a finite set of at least two prime numbers, and A the set of positive integers that are products of powers of primes from P. Let F(k) denote the smallest positive integer which cannot be presented as sum of less than k terms of A.…

Let $H$ be a planar graph. By a classical result of Robertson and Seymour, there is a function $f:\mathbb{N} \to \mathbb{R}$ such that for all $k \in \mathbb{N}$ and all graphs $G$, either $G$ contains $k$ vertex-disjoint subgraphs each…

Let $\mathbb{Z}^{+}$ be the set of positive integers. Let $C_{k}$ denote all subsets of $\mathbb{Z}^{+}$ such that neither of them contains $k + 1$ pairwise coprime integers and $C_k(n)=C_k\cap \{1,2,\ldots,n\}$. Let $f(n, k) =…

One of the central problems in additive combinatorics is to determine how large a subset of the first $N$ integers can be before it is forced to contain $k$ elements forming an arithmetic progression. Around 25 years ago, Gowers proved the…

We show that the difference between consecutive terms in sequences of integers whose greatest prime factor grows slowly tends to infinity.

In 1965 Erd\H{o}s asked, what is the largest size of a family of $k$-element subsets of an $n$-element set that does not have a matching of size $s+1$? In this note, we improve upon a recent result of Frankl and resolve this problem for…

Let $k\geq2$ and $s$ be positive integers. Let $\theta\in(0,1)$ be a real number. In this paper, we establish that if $s>k(k+1)$ and $\theta>0.55$, then every sufficiently large natural number $n$, subjects to certain congruence conditions,…

Our main result is a new upper bound for the size of k-uniform, L-intersecting families of sets, where L contains only positive integers. We characterize extremal families in this setting. Our proof is based on the Ray-Chaudhuri--Wilson…

Erd\H{o}s, Fajtlowicz and Staton asked for the least integer $f(k)$ such that every graph with more than $f(k)$ vertices has an induced regular subgraph with at least $k$ vertices. Here we consider the following relaxed notions. Let $g(k)$…

On the space of rhythms of arbitrary length with a fixed number of onsets, a self map $F$ is constructed. It is shown that for any rhythm $\mathbf{r}$ of the space there exists a nonnegative integer $k$ such that $F^k(\mathbf{r})$ falls…

In a recent work Friedlander studied the problem of how large consecutive prime gaps should be in order that the sum of the reciprocals should be divergent. Supposing a very deep Hypothesis, a generalization of the Hardy--Littlewood prime…

We prove that there are infinitely many $n$ such that $\omega(n+k) \ll \log k$ for all integers $k \ge 2$. This improves on a result of Tao-Ter\"{a}v\"{a}inen (2025), who has $O(k)$ in place of $O(\log k)$. As corollaries, we make progress…

In this paper, we study the linear independence between the distribution of the number of prime factors of integers and that of the largest prime factors of integers. Respectively, under a restriction on the largest prime factors of…