Sums of Palindromes: an Approach via Automata

TL;DR

This paper uses automata-based decision procedures to prove that every natural number can be expressed as a sum of a small number of palindromic numbers in bases 2, 3, and 4, completing previous unresolved cases.

Contribution

It introduces an automata-based decision method to resolve the minimal number of palindromic summands needed in bases 2, 3, and 4, fully solving the problem.

Findings

Every number is sum of at most 4 base-2 palindromes (optimal)

Complete resolution for bases 2, 3, and 4

Automata methods can automate complex number decomposition proofs

Abstract

Recently, Cilleruelo, Luca, & Baxter proved, for all bases b >= 5, that every natural number is the sum of at most 3 natural numbers whose base-b representation is a palindrome. However, the cases b = 2, 3, 4 were left unresolved. We prove, using a decision procedure based on automata, that every natural number is the sum of at most 4 natural numbers whose base-2 representation is a palindrome. Here the constant 4 is optimal. We obtain similar results for bases 3 and 4, thus completely resolving the problem. We consider some other variations on this problem, and prove similar results. We argue that heavily case-based proofs are a good signal that a decision procedure may help to automate the proof.