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Additive Number Theory via Automata Theory

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Abstract

We show how some problems in additive number theory can be attacked in a novel way, using techniques from the theory of finite automata. We start by recalling the relationship between first-order logic and finite automata, and use this relationship to solve several problems involving sums of numbers defined by their base-2 and Fibonacci representations. Next, we turn to harder results. 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 (Cilleruelo et al., Math. Comput. 87, 3023–3055, 2018). However, the cases b = 2, 3, 4 were left unresolved. We prove 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 of palindromes as an additive basis. 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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Acknowledgments

We thank Dirk Nowotka, Parthasarathy Madhusudan, and Jean-Paul Allouche for helpful discussions. We thank the creators of the ULTIMATE automaton library for their assistance. We are also grateful to the referees for reading our paper with care and making several helpful suggestions.

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  1. School of Computer Science, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Aayush Rajasekaran, Jeffrey Shallit & Tim Smith

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  1. Aayush Rajasekaran
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Rajasekaran, A., Shallit, J. & Smith, T. Additive Number Theory via Automata Theory. Theory Comput Syst 64, 542–567 (2020). https://doi.org/10.1007/s00224-019-09929-9

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