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 imply the theorem can be embedded in decidable theories, and show how automated methods can be used to search for these stronger lemmas.
@InProceedings{madhusudan_et_al:LIPIcs.MFCS.2018.18, author = {Madhusudan, P. and Nowotka, Dirk and Rajasekaran, Aayush and Shallit, Jeffrey}, title = {{Lagrange's Theorem for Binary Squares}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, pages = {18:1--18:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-086-6}, ISSN = {1868-8969}, year = {2018}, volume = {117}, editor = {Potapov, Igor and Spirakis, Paul and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.18}, URN = {urn:nbn:de:0030-drops-96003}, doi = {10.4230/LIPIcs.MFCS.2018.18}, annote = {Keywords: binary square, theorem-proving, finite automaton, decision procedure, decidable theory, additive number theory} }
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