Lagrange's Theorem for Binary Squares

@article{Madhusudan2018LagrangesTF,
  title={Lagrange's Theorem for Binary Squares},
  author={P. Madhusudan and Dirk Nowotka and Aayush Rajasekaran and Jeffrey Shallit},
  journal={ArXiv},
  year={2018},
  volume={abs/1710.04247}
}
We show how to prove theorems in additive number theory using a decision procedure based on finite automata. Among other things, we obtain the following analogue of Lagrange's theorem: every natural number > 686 is the sum of at most 4 natural numbers whose canonical base-2 representation is a binary square, that is, a string of the form xx for some block of bits x. Here the number 4 is optimal. While we cannot embed this theorem itself in a decidable theory, we show that stronger lemmas that… 
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