# 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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• 2019
It is shown that for each integer \$k \geq 2\$ there exists a positive integer \$W(k)\$ such that every sufficiently large multiple of \$E_k := \gcd(2^k - 1, k)\$ is the sum of at most the binary \$k\$'th powers.
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• 2018
It is shown that the algorithm to determine the smallestinline-formula content-type, which forms an additive basis for the natural numbers, is effective.
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