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Using Automata Theory to Solve Problems in Additive Number Theory

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Rajasekaran_Aaayush.pdf (373.33 KB)

Date

2018-04-30

Authors

Rajasekaran, Aayush

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University of Waterloo

Abstract

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 whose base-k representations have certain interesting proper- ties. In particular, we look at palindromes, which are numbers whose base-k representations read the same forward and backward, and binary squares, which are numbers whose binary representation is some block repeated twice (like (36)_2 = 100100). We show that all natural numbers are the sum of four binary palindromes. We also show that all natural numbers are the sum of three base-3 palindromes, and are also the sum of three base-4 palindromes. We also show that every sufficiently large natural number is the sum of four binary squares. We establish these results using virtually no number theory at all. Instead, we construct automated proofs using automata. The general proof technique is to build an appropriate machine, and then run decision algorithms to establish our theorems.

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Keywords

Automata Theory, Number Theory, Formal languages, Automated proofs

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http://hdl.handle.net/10012/13202

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Theses
Computer Science