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arXiv · 1706.10206

Sums of Palindromes: an Approach via Automata

Aayush Rajasekaran·Jeffrey Shallit·Tim Smith
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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.

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Aayush Rajasekaran, Jeffrey Shallit, Tim Smith. 2017-06-30. Sums of Palindromes: an Approach via Automata. https://arxiv.org/abs/1706.10206

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