A Problem Based Journey from Elementary Number Theory to an Introduction to Matrix Theory: The President Problems

163 Pages·2021·36.286 MB·English

by  Abraham Berman

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Other World Scientific Titles by the Author
Copositive and Completely Positive Matrices
ISBN: 978-981-120-434-0
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A PROBLEM BASED JOURNEY FROM ELEMENTARY NUMBER THEORY
TO AN INTRODUCTION TO MATRIX THEORY
The President Problems
Copyright © 2022 by World Scientific Publishing Co. Pte. Ltd.
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ISBN 978-981-123-488-0 (ebook for institutions)
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To my grandchildren who enjoy playing with problems
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Contents
Introduction xi
Introduction to the Course . . . . . . . . . . . . . . . . . . . . . . . . xi
1 Algebraic Structures 1
1.1 Groups, Fields and Rings . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5.1 Prestigious prizes in mathematics . . . . . . . . . . . . . . 7
2 The Natural Numbers 11
2.1 What is Induction? . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Mathematical Induction . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Easy to State Open Problems . . . . . . . . . . . . . . . . . . . . 16
2.4 Tiling and Geometry Problems . . . . . . . . . . . . . . . . . . . 17
2.5 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.6 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.7.1 Fermat’s Last Theorem . . . . . . . . . . . . . . . . . . . 26
2.7.2 The Catalan Conjecture . . . . . . . . . . . . . . . . . . . 26
2.7.3 Euler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.7.4 Euclid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.7.5 Gauss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.7.6 Newton and Leibniz . . . . . . . . . . . . . . . . . . . . . 28
2.7.7 Collatz and Tau . . . . . . . . . . . . . . . . . . . . . . . 30
2.7.8 Sylvester–Gallai theorem . . . . . . . . . . . . . . . . . . 30
3 The Integers 31
3.1 The Greatest Common Divisor . . . . . . . . . . . . . . . . . . . 31
3.2 Congruence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.4 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
vii
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viii THE PRESIDENT PROBLEMS
4 The Real Numbers 41
4.1 Sequences and Rational Numbers . . . . . . . . . . . . . . . . . . 41
4.2 Irrational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.3 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.4 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5 Introduction to Set Theory 49
5.1 Countable Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.2 Uncountable Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.3 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.4 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.5.1 Cantor, Fraenkel, Russel and Zermelo . . . . . . . . . . . 59
5.5.2 Hilbert’s 23 Problems . . . . . . . . . . . . . . . . . . . . 59
5.5.3 Gödel and Cohen . . . . . . . . . . . . . . . . . . . . . . . 61
5.5.4 Bernstein and Schröder . . . . . . . . . . . . . . . . . . . 61
6 The Pigeonhole Principle and the Base 2 Number System 63
6.1 The Pigeonhole Principle . . . . . . . . . . . . . . . . . . . . . . 63
6.2 The Base 2 Number System . . . . . . . . . . . . . . . . . . . . . 63
6.3 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6.4 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
6.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
6.5.1 Dirichlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
6.5.2 Ask Marilyn . . . . . . . . . . . . . . . . . . . . . . . . . 68
6.5.3 The Erdős–Szekeres Theorem . . . . . . . . . . . . . . . . 68
7 Introduction to Group Theory 71
7.1 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
7.2 Lagrange’s, Euler’s and Fermat’s Theorems . . . . . . . . . . . . 73
7.3 The RSA Public Key Cybersystem . . . . . . . . . . . . . . . . . 76
7.4 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
7.5 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
7.6 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
8 Introduction to Matrix Theory 87
8.1 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
8.2 Graphs and Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 101
8.3 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
8.4 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
9 Fibonacci Numbers, Determinants and Eigenvalues 109
9.1 The Fibonacci Sequence . . . . . . . . . . . . . . . . . . . . . . . 109
9.2 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
9.3 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . 117
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CONTENTS ix
9.4 The Zeckendorf Representation of the Natural Numbers . . . . . 121
9.5 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
9.6 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
9.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
9.7.1 Fibonacci . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
9.7.2 The golden ratio . . . . . . . . . . . . . . . . . . . . . . . 130
9.7.3 Cayley and Hamilton. . . . . . . . . . . . . . . . . . . . . 131
9.7.4 The Friendship Theorem. . . . . . . . . . . . . . . . . . . 131
10 The Mathematics Behind Google’s Page Rank
and a Game of Numbers 133
10.1 Page Rank. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
10.2 Back to the Numbers on the Pentagon Problem . . . . . . . . . . 140
10.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
10.3.1 Perron and Frobenius . . . . . . . . . . . . . . . . . . . . 141
10.3.2 Brin and Page . . . . . . . . . . . . . . . . . . . . . . . . 141
10.3.3 Alon, Peres, Mozes and Eriksson . . . . . . . . . . . . . . 141
Bibliography 145
Index 147