Note on a theorem of Birch and Erdős

Yuchen Ding, Honghu Liu

Published 2025-03-01, updated 2025-05-04Version 2

Let $p,q>1$ be two relatively prime integers and $\mathbb{N}$ the set of nonnegative integers. Let $f_{p,q}(n)$ be the number of different expressions of $n$ written as a sum of distinct terms taken from $\{p^{\alpha}q^{\beta}:\alpha,\beta\in \mathbb{N}\}$. Erd\H os conjectured and then Birch proved that $f_{p,q}(n)\ge 1$ provided that $n$ is sufficiently large. In this note, for all sufficiently large number $n$ we prove $$ f_{p,q}(n)=2^{\frac{(\log n)^2}{2\log p\log q}\big(1+O(\log\log n/\log n)\big)}. $$ We also prove that $f_{2,q}(n+1)\ge f_{2,q}(n)$ for any $n\ge 1$ and furthermore classify all the situations of $n$ such that $f_{2,q}(n+1)=f_{2,q}(n)$. Additionally, we will show that $$ \lim_{n\rightarrow\infty}f_{2,q}(n+1)/f_{2,q}(n)=1. $$