Note on a theorem of Birch and Erdős

Abstract

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 $\{p^{\hat{I}}\pm q^2:\hat{I}\pm ,{\hat{I}}^2\in \mathbb{N}\}$. 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 $$f_{p,q}(n)=2^{\frac{(\log n{)}^2}{2\log p\log q}(1+O(\log \log n/\log n))}.$$ We also show that ${\lim }_{n\to \infty }{f}_{2,q}(n+1)/f_{2,q}(n)=1.$ Additionally, we will point out the relations between $f_{2,q}(n)$ and $m$-ary partitions.

some proofs are adjusted