Niven numbers are an asymptotic basis of order 3

TL;DR

The paper proves that for any base g ≥ 3, all large enough natural numbers can be expressed as the sum of three Niven numbers, providing an asymptotic count of such representations.

Contribution

It establishes that Niven numbers form an asymptotic basis of order 3 for all bases g ≥ 3, with an asymptotic formula for their sum representations.

Findings

Every large number is a sum of three Niven numbers for g ≥ 3

Provides an asymptotic formula for the number of representations

Shows Niven numbers form an asymptotic basis of order 3

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.