Niven numbers are an asymptotic basis of order 3

TL;DR

This paper resolves the question of whether base-$g$ Niven numbers form an asymptotic basis of order $3$ for every base $g\ge 3$ by deploying the circle method to count representations of large integers as sums of three near-average-digit-sum integers, and then showing a positive fraction of these representations are Niven numbers. The authors develop a probabilistic digit model, a centred base-$g$ expansion, and a local limit theorem to control major- and minor-arc contributions, yielding an explicit main-term formula for representations $r_{S_1+S_2+S_3}(M)$ and, under suitable constraints on the fixed digit sums $k_1,k_2,k_3$, an asymptotic count $r_{\mathcal N_1+\mathcal N_2+\mathcal N_3}(M)$. With suitable gcd and congruence conditions, these counts translate into a similar asymptotic formula for sums of three base-$g$ Niven numbers, establishing the unconditional order-3 basis property. The results advance the understanding of digit-restricted additive bases and quantify the combinatorial richness of Niven-number representations in base $g$.

Abstract

A base-$g$ Niven number is a natural number divisible by the sum of its base-$g$ digits. We show that, for any $g\geq 3$, all sufficiently large natural numbers can be written as the sum of three base-$g$ Niven numbers. We also give an asymptotic formula for the number of representations of a sufficiently large integer as the sum of three integers with fixed, close to average, digit sums.

Theorems & Definitions (53)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 2.1
• Proposition 2.2
• Proposition 2.3
• proof : Proof of \ref{['Main theorem S count']}
• Definition 2.4
• Corollary 3.1
• proof