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Sums of Palindromes: an Approach via Automata

Authors Aayush Rajasekaran, Jeffrey Shallit, Tim Smith

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Aayush Rajasekaran
Jeffrey Shallit
Tim Smith

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Aayush Rajasekaran, Jeffrey Shallit, and Tim Smith. Sums of Palindromes: an Approach via Automata. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 54:1-54:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.STACS.2018.54

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.

Subject Classification

Keywords
  • finite automaton
  • nested-word automaton
  • decision procedure
  • palindrome
  • additive number theory

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