Note on a theorem of Birch and Erdős

2025/03/01 by Ding, Yuchen, Liu, Honghu, Wang, Zi
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2503.11676

Abstract

Let p,q>1 be two relatively prime integers and 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∈ ℕ\. Erd\H os conjectured and then Birch proved that f_p,q(n)≥ 1 provided that n is sufficiently large. In this note, for all sufficiently large number n we prove f_p,q(n)=2^{((log n)2)/(2log plog q)(1+O(loglog n/log n))}. We also show that lim_n→∞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.

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