Sums of Palindromes: an Approach via Automata
Formal Languages and Automata Theory
2017-09-01 v3 Combinatorics
Number Theory
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.
Keywords
Cite
@article{arxiv.1706.10206,
title = {Sums of Palindromes: an Approach via Automata},
author = {Aayush Rajasekaran and Jeffrey Shallit and Tim Smith},
journal= {arXiv preprint arXiv:1706.10206},
year = {2017}
}