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.

中文翻译：

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基于自动机理论的加法数论

我们展示了如何使用有限自动机理论的技术，以新颖的方式解决加数理论中的某些问题。我们首先回顾一下一阶逻辑与有限自动机之间的关系，并使用此关系来解决一些问题，这些问题涉及以基数2和斐波那契表示形式定义的数字之和。接下来，我们转向更困难的结果。最近，Cilleruelo，卢卡，＆巴克斯特证明，对于所有碱基b ≥5，每一个自然数至多为3的自然数，其碱基的总和b表示是回文（Cilleruelo等，数学COMPUT。87， 3023–3055，2018年）。但是，情况b= 2，3，4未解决。我们证明每个自然数是最多4个自然数之和，其2表示形式是回文。在这里，常数4是最佳的。我们对碱基3和4获得了相似的结果，从而完全解决了回文问题作为加法基础的问题。我们考虑此问题的其他变体，并证明相似的结果。我们认为，大量基于案例的证明是一个很好的信号，表明决策程序可能有助于自动化证明。