A quantitative form of the Erdős–Birch theorem
Volume 178 / 2017
Acta Arithmetica 178 (2017), 301-311
MSC: 11A07, 11B13.
DOI: 10.4064/aa8434-10-2016
Published online: 10 May 2017
Abstract
In 1959, B. J. Birch proved that for any coprime integers $p,q$ greater than 1, there exists a number $B$ such that every integer $n \gt B$ can be expressed as the sum of distinct terms taken from $\{ p^aq^b \mid a\ge 0,\, b\ge 0, a, b\in \mathbb{Z}\} $. In this paper, it is proved that there exist two positive integers $K$ and $B$ with $\log_2 \log_2 K \lt q^{2p}$ and $\log_2 \log_2 \log_2 B \lt q^{2p}$ such that every integer $n\ge B$ can be expressed as the sum of distinct terms taken from $\{p^aq^b \mid a\ge 0,\, 0\le b\le K,\, a+b \gt 0,\, a, b\in \mathbb{Z}\}$, where $\log_2$ means the logarithm to base 2. Up to our knowledge, this is the first bound for $B$.
