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ALMOST ALL PRIMES HAVE A MULTIPLE OF SMALL HAMMING WEIGHT

Part of: Elementary number theory Sequences and sets Additive number theory; partitions Multiplicative number theory

Published online by Cambridge University Press:  23 May 2016

CHRISTIAN ELSHOLTZ
CHRISTIAN ELSHOLTZ*
Affiliation:
Institute of Analysis and Number Theory, Graz University of Technology, Kopernikusgasse 24, A-8010 Graz, Austria email elsholtz@math.tugraz.at
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Abstract

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We improve recent results of Bourgain and Shparlinski to show that, for almost all primes $p$, there is a multiple $mp$ that can be written in binary as

$$\begin{eqnarray}mp=1+2^{m_{1}}+\cdots +2^{m_{k}},\quad 1\leq m_{1}<\cdots <m_{k},\end{eqnarray}$$
with $k=6$ (corresponding to Hamming weight seven). We also prove that there are infinitely many primes $p$ with a multiplicative subgroup $A=\langle g\rangle \subset \mathbb{F}_{p}^{\ast }$, for some $g\in \{2,3,5\}$, of size $|A|\gg p/(\log p)^{3}$, where the sum–product set $A\cdot A+A\cdot A$ does not cover $\mathbb{F}_{p}$ completely.

Keywords

primesadditive basessumsetsdistribution of integers with multiplicative constraintsWaring’s problem and variants

MSC classification

Primary: 11N25: Distribution of integers with specified multiplicative constraints
Secondary: 11A41: Primes 11A63: Radix representation; digital problems 11B13: Additive bases, including sumsets 11B50: Sequences (mod $m$) 11P05: Waring's problem and variants
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 94 , Issue 2 , October 2016 , pp. 224 - 235
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

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