Nuggets of Number Theory: A Visual Approach

Cover......Page 1
Title page......Page 4
Copyright......Page 5
Contents......Page 6
Preface......Page 10
1.1. Polygonal numbers......Page 12
1.2. Triangular number identities......Page 18
1.3. Oblong numbers and the infinitude of primes......Page 24
1.4. Pentagonal and other figurate numbers......Page 25
1.5. Polite numbers......Page 27
1.6. Three-dimensional figurate numbers......Page 29
1.7. Exercises......Page 32
2.1. Congruence results for triangular numbers......Page 36
2.2. Congruence results for other figurate numbers......Page 38
2.3. Fermat’s little theorem......Page 41
2.4. Wilson’s theorem......Page 43
2.5. Exercises......Page 44
Chapter 3. Diophantine Equations......Page 46
3.1. Triangles and squares......Page 47
3.2. Linear Diophantine equations......Page 49
3.3. Linear congruences and the Chinese remainder theorem......Page 52
3.4. The Pell equation ²-2²=1......Page 55
3.5. The Pell equation ²-3²=1......Page 57
3.6. The Pell equations ²-²=1......Page 60
3.7. Exercises......Page 63
Chapter 4. Pythagorean Triples......Page 66
4.1. Euclid’s formula......Page 67
4.2. Pythagorean triples and means of odd squares......Page 68
4.3. The carpets theorem......Page 69
4.4. Pythagorean triples and the factors of even squares......Page 70
4.5. Almost isosceles primitive Pythagorean triples......Page 72
4.6. A Pythagorean triple tree......Page 75
4.8. Pythagorean primes and triangular numbers......Page 78
4.9. Divisibility properties......Page 80
4.10. Pythagorean triangles......Page 81
4.12. Sums of two squares......Page 84
4.13. Pythagorean quadruples and Pythagorean boxes......Page 87
4.14. Exercises......Page 90
5.1. The irrationality of √2......Page 94
5.2. Rational approximations to √2: Pell equations......Page 99
5.3. Rational approximations to √2: Almost isosceles PPTs......Page 100
5.4. The irrationality of √3 and √5......Page 101
5.5. The irrationality of √ for non-square ......Page 104
5.6. The golden ratio and the golden rectangle......Page 105
5.7. The golden ratio and the regular pentagon......Page 107
5.8. Periodic continued fractions......Page 109
5.9. Exercises......Page 112
Chapter 6. Fibonacci and Lucas Numbers......Page 114
6.1. The Fibonacci sequence in art and nature......Page 115
6.2. Fibonacci parallelograms, triangles, and trapezoids......Page 117
6.3. Fibonacci rectangles and squares......Page 118
6.4. Diagonal sums in Pascal’s triangle......Page 124
6.5. Lucas numbers......Page 126
6.6. The Pell equations ²-5²=±4 and Binet’s formula......Page 128
6.7. Exercises......Page 132
7.1. Euclid’s formula......Page 134
7.2. Even perfect numbers and geometric progressions......Page 136
7.3. Even perfect numbers and triangular numbers......Page 137
7.5. Even perfect numbers end in 6 or 28......Page 139
7.6. Even perfect numbers modulo 7......Page 141
7.8. Odd perfect numbers......Page 142
7.9. Exercises......Page 144
Chapter 1......Page 146
Chapter 2......Page 148
Chapter 3......Page 149
Chapter 4......Page 152
Chapter 5......Page 153
Chapter 6......Page 155
Chapter 7......Page 156
Bibliography......Page 160
Index......Page 162
Back Cover......Page 165