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Number Theory Revealed: The Series
by Andrew Granville
Andrew Granville

There will be four books in this trilogy:

  1. An elementary book in two versions: Number Theory Revealed: An Introduction and Number Theory Revealed: A Masterclass (published in 2019).
  2. Followed by an introduction to analytic number theory: Analytic number theory revealed: The distribution of prime numbers.
  3. Complemented by an introduction to solutions of equations of low degree: Arithmetic Geometry Revealed: Rational Points on Curves, and Modular Forms.
  4. And finally, the author will revisit one of the great classics, Gauss's Disquisitiones Arithmeticae Revealed.

Number Theory Revealed: An Introduction and Number Theory Revealed: A Masterclass

The majority of these links are pdf files that reside on the author's personal website
Preface: General Preface
Why study number theory?. Why give proofs?. Motivation and expectations
Homework. Further exploration. The cover. Gauss's Disquisitiones Arithmeticae
The language of mathematics.
Chapter 0: Preliminary Chapter on Induction.
Appendix 0A: A closed formula for sums of powers
Homework: Problems and Hints
Chapter 1: The Euclidean Algorithm.
Appendix 1A: Reformulating the Euclidean Algorithm
Homework: Problems and Hints
Chapter 2: Congruences
Appendix 2A: Congruences in the language of groups
Homework: Problems and Hints
Chapter 3: The basic algebra of number theory
Appendix 3A: Factoring binomial coefficients, and Pascal's Triangle mod p
Homework: Problems and Hints
Chapter 4: Multiplicative functions
Appendix 4C: Irreducible polynomials mod p
Homework: Problems and Hints
Chapter 5: The Distribution of Prime Numbers
Appendix 5A: Bertrand's Postulate and beyond
Homework: Problems and Hints
Conway's Prime producing machine (Details in section 5.20). Search for Ulam's spiral.
Chapter 6: Diophantine problems
Appendix 6A: Polynomial solutions of Diophantine Equations
Homework: Problems and Hints
Chapter 7: Power Residues
Appendix 7A: Card Shuffling and Fermat's Little Theorem
Homework: Problems and Hints
Chapter 8: Quadratic residues
Appendix 8A: Eisenstein's proof of quadratic reciprocity
Homework: Problems and Hints
Chapter 9: Quadratic equations
Appendix 9D: Descent and the quadratics
Homework: Problems and Hints
Chapter 10: Square Roots and Factoring
Appendix 10A: Pseudoprime tests using square roots of 1
Homework: Problems and Hints
Chapter 11: Rational approximations to real numbers
Appendix 11A: Uniform distribution
Homework: Problems and Hints
Chapter 12: Binary quadratic forms
Appendix 12A: Composition rules: Gauss, Dirichlet and Bhargava
Homework: Problems and Hints
Chapter 13: The anatomy of integers
Appendix 13A: Other anatomies
Homework: Problems and Hints
Chapter 14: Counting integral and rational points on curves, mod p
Appendix 14A: Gauss sums
Homework: Problems and Hints
Chapter 15: Combinatorial number theory
Appendix 15A: Summing sets (mod p)
Homework: Problems and Hints
Chapter 16: The p-adic numbers
Chapter 16.7: The p-adic dilogarithm
Homework: Problems and Hints
Chapter 17: Rational points on elliptic curves
Appendix 17A: General Mordell's Theorem
Homework: Problems and Hints
References: The Great Books of number theory.
Recommended Further Reading