A quantitative form of the Erdős–Birch theorem

Jin-Hui Fang¹ ; Yong-Gao Chen²

¹ Department of Mathematics Nanjing University of Information Science and Technology Nanjing 210044, P.R. China
² School of Mathematical Sciences and Institute of Mathematics Nanjing Normal University Nanjing 210023, P.R. China

Acta Arithmetica, Tome 178 (2017) no. 4, pp. 301-311.

Résumé

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 $B$.

Détail

DOI : 10.4064/aa8434-10-2016

Mots-clés : birch proved coprime integers greater nbsp there exists number every integer expressed sum distinct terms taken mid mathbb paper proved there exist positive integers log log log log log every integer expressed sum distinct terms taken mid mathbb where log means logarithm base nbsp knowledge first bound