ALMOST ALL PRIMES HAVE A MULTIPLE OF SMALL HAMMING WEIGHT

CHRISTIAN ELSHOLTZ

doi:10.1017/S000497271600023X

ALMOST ALL PRIMES HAVE A MULTIPLE OF SMALL HAMMING WEIGHT

CHRISTIAN ELSHOLTZ^*

^*Corresponding author for this work

Institute of Analysis and Number Theory (5010)

Research output: Contribution to journal › Article › peer-review

Abstract

We improve recent results of Bourgain and Shparlinski to show that, for almost all primes (Formula presented.), there is a multiple (Formula presented.) that can be written in binary as (Formula presented.) with (Formula presented.) (corresponding to Hamming weight seven). We also prove that there are infinitely many primes (Formula presented.) with a multiplicative subgroup (Formula presented.), for some (Formula presented.), of size (Formula presented.), where the sum–product set (Formula presented.) does not cover (Formula presented.) completely.

Original language	English
Pages (from-to)	224-235
Number of pages	12
Journal	Bulletin of the Australian Mathematical Society
Volume	94
Issue number	2
DOIs	https://doi.org/10.1017/S000497271600023X
Publication status	Published - 23 May 2016

Keywords

additive bases
distribution of integers with multiplicative constraints
primes
sumsets
Waring’s problem and variants

ASJC Scopus subject areas

General Mathematics

Fields of Expertise

Information, Communication & Computing

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10.1017/S000497271600023X

Cite this

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AB - We improve recent results of Bourgain and Shparlinski to show that, for almost all primes (Formula presented.), there is a multiple (Formula presented.) that can be written in binary as (Formula presented.) with (Formula presented.) (corresponding to Hamming weight seven). We also prove that there are infinitely many primes (Formula presented.) with a multiplicative subgroup (Formula presented.), for some (Formula presented.), of size (Formula presented.), where the sum–product set (Formula presented.) does not cover (Formula presented.) completely.

KW - additive bases

KW - distribution of integers with multiplicative constraints

KW - primes

KW - sumsets

KW - Waring’s problem and variants

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ALMOST ALL PRIMES HAVE A MULTIPLE OF SMALL HAMMING WEIGHT

Abstract

Keywords

ASJC Scopus subject areas

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