Using Automata Theory to Solve Problems in Additive Number Theory

Automata Theory;Number Theory;Formal languages;Automated proofs

Rajasekaran, Aayushadvisor:Shallit, Jeffrey ; affiliation1:Faculty of Mathematics ; Shallit, Jeffrey ;

University of Waterloo

关键词: Master Thesis;    Formal languages;    Automated proofs;    Number Theory;    Automata Theory;   

Others  :  https://uwspace.uwaterloo.ca/bitstream/10012/13202/3/Rajasekaran_Aaayush.pdf

瑞士|英语

来源: UWSPACE Waterloo Institutional Repository

【 摘 要 】

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.