Lagrange's Theorem for Binary Squares

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…
Using Automata Theory to Solve Problems in Additive Number Theory
Additive number theory is the study of the additive properties of integers. Perhaps the best-known theorem is Lagrange’s result that every natural number is the sum of four squares. We study numbers
Additive Number Theory via Automata Theory
• Mathematics, Computer Science
Theory of Computing Systems
• 2019
It is shown how some problems in additive number theory can be attacked in a novel way, using techniques from the theory of finite automata, and it is argued that heavily case-based proofs are a good signal that a decision procedure may help to automate the proof.
A method of producing results in additive number theory, relying on theoremproving software and an approximation technique is introduced, which proves that every natural number greater than 25 can be written as the sum of at most 3 natural numbers whose canonical base-2 representations have an equal number of 0's and 1’s.
Waring's Theorem for Binary Powers
• Mathematics, Computer Science
Comb.
• 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.
When is an automatic set an additive basis?
• Mathematics, Computer Science
Proceedings of the American Mathematical Society, Series B
• 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.