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ALMOST ALL PRIMES HAVE A MULTIPLE OF SMALL HAMMING WEIGHT
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 94 / Issue 2 / October 2016
- Published online by Cambridge University Press:
- 23 May 2016, pp. 224-235
- Print publication:
- October 2016
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- Article
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- You have access
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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 with
$$\begin{eqnarray}mp=1+2^{m_{1}}+\cdots +2^{m_{k}},\quad 1\leq m_{1}<\cdots <m_{k},\end{eqnarray}$$
$k=6$ (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.
