| Preface:
General Preface
Why study number theory?.
Why give proofs?.
Motivation and expectations
Homework.
Special Features.
Further exploration.
The cover.
Gauss's Disquisitiones Arithmeticae
The language of mathematics.
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| Chapter 0:
Preliminary Chapter on Induction. Appendix 0A:
A closed formula for sums of powers
Homework: Problems and Hints
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| Chapter 1:
The Euclidean Algorithm. Appendix 1A:
Reformulating the Euclidean Algorithm
Homework: Problems and Hints
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| Chapter 2:
Congruences Appendix 2A:
Congruences in the language of groups
Homework: Problems and Hints
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| Chapter 3:
The basic algebra of number theory Appendix 3A:
Factoring binomial coefficients and Pascal's Triangle mod p
Homework: Problems and Hints
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| Chapter 4:
Multiplicative functions Appendix 4C:
Irreducible polynomials mod p
Homework: Problems and Hints
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| Chapter 5:
The Distribution of Prime Numbers Appendix 5A:
Bertrand's Postulate and beyond
Bonus Read: A Review of Prime problems
Homework: Problems and Hints
Conway's Prime producing machine
(Details in section 5.22).
Ulam's spiral |
| Chapter 6:
Diophantine problems Appendix 6A:
Polynomial solutions of Diophantine Equations
Homework: Problems and Hints
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| Chapter 7:
Power Residues Appendix 7A:
Card Shuffling and Fermat's Little Theorem
Homework: Problems and Hints
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| Chapter 8:
Quadratic residues Appendix 8A:
Eisenstein's proof of quadratic reciprocity
Matt Baker's new card shuffling proof and A great online explanation
Homework: Problems and Hints
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| Chapter 9:
Quadratic equations Appendix 9D:
Descent and the quadratics
Homework: Problems and Hints
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| Chapter 10:
Square Roots and Factoring Appendix 10A:
Pseudoprime tests using square roots of 1
Homework: Problems and Hints
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| Chapter 11:
Rational approximations to real numbers Appendix 11A:
Uniform distribution
Homework: Problems and Hints
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| Chapter 12:
Binary quadratic forms Appendix 12A:
Composition rules: Gauss, Dirichlet and Bhargava
Homework: Problems and Hints
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| Chapter 13:
The anatomy of integers Appendix 13A:
Other anatomies
Homework: Problems and Hints
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| Chapter 14:
Counting integral and rational points on curves, mod p Appendix 14A:
Gauss sums
Homework: Problems and (No) Hints
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| Chapter 15:
Combinatorial number theory Appendix 15A:
Summing sets modulo p
Homework: Problems and Hints
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| Chapter 16:
The p-adic numbers Chapters 16.6 and 16.7:
The p-adic logarithm and dilogarithm
Homework: Problems and Hints
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| Chapter 17:
Rational points on elliptic curves Appendix 17B:
Pythagorean triangles of area 6
Homework: Problems and (No) Hints
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| Errors and Ideas
The Errata
Ideas For the next edition
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| References:
The
Great Books of number theory.
Recommended Further Reading
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| Blogs:
In each chapter a blog for profs to share their favourite question or explanation
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