A Cascade of Numbers An Introduction to Number Theory
by Burn, Bob; Chetwynd, AmandaFormat: Paperback
Pub. Date: 1995-12-22
Publisher(s): Hodder Education Publishers
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Summary
Preface
Definitions
Part I
1. The magic of nines (decimal place value)
2. Back to basics (bicimal place value)
3. Children's ages (prime factorisation part i)
4. The prison door
Table of Contents
| Preface | |
| Definitions | |
| The magic of nines (decimal place value) | p. 3 |
| Back to basics (bicimal place value) | p. 5 |
| Children's ages (prime factorisation part i) | p. 5 |
| The prison door problem (counting factors) | p. 5 |
| Catching practice (repeated addition around a circle part i) | p. 7 |
| Algorithma (highest common factor, part i) | p. 8 |
| Breeding rabbits (Fibonacci, using factors) | p. 10 |
| To divide or not to divide (prime factorisation part ii) | p. 12 |
| A return visit to Algorithma (highest common factor, part ii, the fundamental theorem of arithmetic) | p. 13 |
| The stamp problem | p. 13 |
| Tests for divisibility (decimal place value) | p. 15 |
| Spot check (first taste of modulo 10) | p. 15 |
| Eratosthenes' sieve (locating primes, proof by contradiction) | p. 17 |
| Raffle tickets and neighbours (primes and composites) | p. 17 |
| How many primes? (no finite number, contradiction again) | p. 18 |
| No shuffling (first taste of modulo 4) | p. 19 |
| Can you really tell the time? (modulo 12 and modulo 4) | p. 19 |
| Fibonacci numbers and the division algorithm | p. 21 |
| Relations can be difficult (equivalence) | p. 22 |
| Dominoes (induction) | p. 25 |
| Chinese remainders | p. 31 |
| Systematic catching practice (repeated addition part ii, [Phi]) | p. 32 |
| Do you know your tables? (modular multiplication) | p. 33 |
| Coding and decoding (simple methods) | p. 34 |
| Repacking (squares and their residues) | p. 35 |
| Where have all the squares gone? | p. 36 |
| Where have all the squares come from? | p. 36 |
| How old is Grandma? (applying quadratic residues) | p. 37 |
| Higher powers (approaching Fermat's theorem) | p. 39 |
| Just shuffling and then [superscript power] (modulo 7) (Fermat's theorem part i) | p. 40 |
| Just shuffling and then [superscript power] (modulo 3, 5, 11) (Fermat's theorem part ii) | p. 41 |
| Factorials (Wilson's theorem) | p. 43 |
| Square roots of -1, prime modulus (p [actual symbol not reproducible] 1, 3 (mod 4)) | p. 43 |
| How many square roots of -1? (factors of x[superscript 2] + 1) | p. 44 |
| Sums of squares | p. 44 |
| Sums of squares in two ways | p. 46 |
| Pythagorean triples | p. 47 |
| Squares and non-squares (quadratic residues and products) | p. 51 |
| Powers of squares and non-squares (towards the Legendre symbol) | p. 52 |
| The frequency of factors [actual symbol not reproducible] | p. 52 |
| Multiplication like addition (cyclic groups and generators) | p. 53 |
| Powers to a prime modulus (primitive roots) | p. 54 |
| Zero products (modulo a non-prime) | p. 56 |
| Non-zero products (Fermat-Euler theorem) | p. 57 |
| Decimals to the death (recurring decimals to fractions) | p. 57 |
| Recurring decimals (fractions to recurring decimals) | p. 58 |
| Can you reveal all and keep it secret? (Public Key System) | p. 60 |
| Primes as squares and non-squares (starting quadratic reciprocity) | p. 62 |
| Counting dots in a rectangle (odd or even) | p. 64 |
| Half-size products (Gauss' lemma, when 2 is a square) | p. 64 |
| From dashes to dots (Eisenstein) | p. 66 |
| Quadratic reciprocity | p. 68 |
| Adding squares (primes in various forms) | p. 70 |
| Index | p. 147 |
| Table of Contents provided by Blackwell. All Rights Reserved. |
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