Represent a natural number as the sum of palindromes in various bases

TL;DR

This paper proves that any natural number can be represented as a sum of a small number of palindromes in any base, establishing a universal additive basis property.

Contribution

It demonstrates that the set of palindromes forms an additive basis for all natural numbers in any base, with a bound proportional to the base's digit length.

Findings

Every natural number can be expressed as the sum of O(d) palindromes in base d.

Palindromes form an additive basis for natural numbers in any base.

The number of palindromes needed is linearly related to the number of digits.

Abstract

It is shown that the set of palindromes is an additive basis for the natural numbers in any base. Specifically, we prove that every natural number can be expressed as the sum of $O(d)$ palindromes in base $d$.