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Lagrange's Theorem for Binary Squares

Authors P. Madhusudan , Dirk Nowotka , Aayush Rajasekaran, Jeffrey Shallit

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Author Details

P. Madhusudan
  • Department of Computer Science, Thomas M. Siebel Center for Computer Science, 201 North Goodwin Avenue, Urbana, IL 61801-2302, USA
Dirk Nowotka
  • Department of Computer Science, Kiel University, D-24098 Kiel, Germany
Aayush Rajasekaran
  • School of Computer Science, University of Waterloo, Waterloo, ON N2L 3G1, Canada
Jeffrey Shallit
  • School of Computer Science, University of Waterloo, Waterloo, ON N2L 3G1, Canada

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P. Madhusudan, Dirk Nowotka, Aayush Rajasekaran, and Jeffrey Shallit. Lagrange's Theorem for Binary Squares. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 18:1-18:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Models of computation
  • Theory of computation → Constructive mathematics
  • Mathematics of computing → Discrete mathematics
  • binary square
  • theorem-proving
  • finite automaton
  • decision procedure
  • decidable theory
  • additive number theory


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