3-tuples have at most 7 prime factors infinitely often

James Maynard

Type: Article

Publication Date: 2013-06-18

Citations: 9

DOI: https://doi.org/10.1017/s0305004113000339

Abstract

Let $L_1$, $L_2$ $L_3$ be integer linear functions with no fixed prime divisor. We show there are infinitely many $n$ for which the product $L_1(n)L_2(n)L_3(n)$ has at most 7 prime factors, improving a result of Porter. We do this by means of a weighted sieve based upon the Diamond-Halberstam-Richert multidimensional sieve.

Locations

Mathematical Proceedings of the Cambridge Philosophical Society - View
arXiv (Cornell University) - View - PDF
Oxford University Research Archive (ORA) (University of Oxford) - View - PDF
DataCite API - View

Similar Works

Action	Title	Year	Authors
+	Almost-prime $k$-tuples	2012	James E. Maynard
+	Almost-prime $k$-tuples	2012	James Maynard
+	The linear sieve	1996	Melvyn B. Nathanson
+	ALMOST‐PRIME ‐TUPLES	2013	James Maynard
+	Variants of the Selberg sieve, and almost prime k-tuples	2022	Paweł Lewulis
+	An elementary sieve	2007	Damián Gulich Gustavo Funes Leopoldo Garavaglia Beatriz Ruíz M. Garavaglia
+	An elementary sieve	2007	Damián Gulich Gustavo Funes Leopoldo Garavaglia Beatriz Ruíz M. Garavaglia
+	Prime Number Sieves	2020	G. Reitsma
+	Representing an integer as the sum of a prime and the product of two small factors	2020	Roger C. Baker Glyn Harman
+	On an almost-prime sieve	2019	Hieu T. Ngo
+	Representing an integer as the sum of a prime and the product of two small factors.	2020	Roger C. Baker Glyn Harman
+ PDF Chat	Almost primes between all squares	2025	Adrian W. Dudek Daniel R. Johnston
+	A sieve problem over the Gaussian integers	2010	D. R. Heath‐Brown
+	A sieve problem over the Gaussian integers	2010	Waldemar Schlackow
+	An Introduction to Prime Number Sieves	1990	Jonathan Sorenson
+	The Sieve of Prime Numbers Using Tables	2014	Barar Stelian Liviu
+	Some Applications of Sieves of Dimension exceeding 1	1997	Harold G. Diamond H. Halberstam
+	A Sieve Method for Factoring Numbers of the Form n 2 + 1	1959	Daniel Shanks
+	A Friendly Intro to Sieves with a Look Towards Recent Progress on the Twin Primes Conjecture	2014	David Lowry-Duda
+	A Friendly Intro to Sieves with a Look Towards Recent Progress on the Twin Primes Conjecture	2014	David Lowry-Duda

Works That Cite This (8)

Action	Title	Year	Authors
+	Almost prime triples and Chen's theorem	2016	D. R. Heath‐Brown Xiannan Li
+ PDF Chat	Triples of almost primes	2023	Jiamin Li Jianya Liu
+ PDF Chat	Groups with bounded centralizer chains and the Borovik–Khukhro conjecture	2018	А. А. Бутурлакин D. O. Revin A. V. Vasil’ev
+	Bounded Gaps Between Primes	2021	Kevin Broughan
+	ALMOST‐PRIME ‐TUPLES	2013	James Maynard
+ PDF Chat	Small gaps between the set of products of at most two primes	2019	Keiju Sono
+ PDF Chat	Variants Of the Selberg Sieve and Almost Prime<i>K</i>-tuples	2022	Paweł Lewulis
+ PDF Chat	Almost primes in various settings	2022	Paweł Lewulis

Works Cited by This (9)

Action	Title	Year	Authors
+ PDF Chat	Some numerical results in the Selberg sieve method	1972	J. W. Porter
+	ON THE REPRESENTATION OF A LARGER EVEN INTEGER AS THE SUM OF A PRIME AND THE PRODUCT OF AT MOST TWO PRIMES	1973	Chen Jing-Run
+	Multiplicative Number Theory	1980	H. Davenport
+	On almost prime k-tuples	2006	Kwan-hung Ho Kai-Man Tsang
+	Almost‐prime <i>k</i> ‐tuples	1997	D. R. Heath‐Brown
+ PDF Chat	An induction principle for the generalization of Bombieri's prime number theorem	1976	Yoichi Motohashi
+	A Higher-Dimensional Sieve Method	2008	Harold G. Diamond H. Halberstam William F. Galway
+	Multiplicative Number Theory I: Classical Theory	2006	Hugh L. Montgomery R. C. Vaughan
+	Some Applications of Sieves of Dimension exceeding 1	1997	Harold G. Diamond H. Halberstam