Note on a theorem of Birch and Erdős
Yuchen Ding, Honghu Liu, Zi Wang
TL;DR
The paper analyzes the counting function $f_{p,q}(n)$, the number of representations of $n$ as sums of distinct terms drawn from $\{p^{\alpha}q^{\beta}:\alpha,\beta\in\mathbb{N}\}$. It establishes a sharp log-square asymptotic $f_{p,q}(n)=2^{\frac{(\log n)^2}{2\log p\log q}\,(1+O(\log\log n/\log n))}$ for all sufficiently large $n$, extending the Birch–Erdős result with a precise growth rate. The paper also proves a combinatorial recurrence for the case $(p,q)=(2,q)$ (with $q$ odd), $f(n+1)=f(n)$ if $q\nmid (n+1)$ and $f(n+1)=f(n)+f((n+1)/q)$ if $q| (n+1)$, and derives parity-density results showing substantial even and odd value frequencies; it further shows $\lim_{n\to\infty} f_{2,q}(n+1)/f_{2,q}(n)=1$ for odd $q$, linking the problem to $m$-ary partitions and Mahler-type recurrences. These results illuminate the deep combinatorial structure behind representations by products of powers and connect to broader partition-theoretic phenomena with implications for related conjectures and asymptotics.
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 $\{p^αq^β:α,β\in \mathbb{N}\}$. 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 show that $\lim_{n\rightarrow\infty}f_{2,q}(n+1)/f_{2,q}(n)=1.$ Additionally, we will point out the relations between $f_{2,q}(n)$ and $m$-ary partitions.