2019;
587 pp;
Hardcover

MSC: Primary 11;

**Print ISBN: 978-1-4704-4158-6
Product Code: MBK/127**

List Price: $99.00

AMS Member Price: $79.20

MAA Member Price: $89.10

**Electronic ISBN: 978-1-4704-5424-1
Product Code: MBK/127.E**

List Price: $99.00

AMS Member Price: $79.20

MAA Member Price: $89.10

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#### Supplemental Materials

# Number Theory Revealed: A Masterclass

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*Andrew Granville*

Number Theory Revealed: A Masterclass presents a
fresh take on congruences, power residues, quadratic residues, primes,
and Diophantine equations and presents hot topics like cryptography,
factoring, and primality testing. Students are also introduced to
beautiful enlightening questions like the structure of Pascal's
triangle mod p, Fermat's Last Theorem for polynomials, and modern
twists on traditional questions.

This Masterclass edition contains many additional chapters
and appendices not found in Number Theory Revealed: An
Introduction. It is ideal for instructors who wish to tailor a
class to their own interests and gives well-prepared students further
opportunities to challenge themselves and push beyond core number
theory concepts, serving as a springboard to many current themes in
mathematics. Additional topics in A Masterclass include the curvature
of circles in a tiling of a circle by circles, the latest discoveries
on gaps between primes, magic squares of primes, a new proof of
Mordell's Theorem for congruent elliptic curves, as well as links with
algebra, analysis, cryptography, and dynamics.

This book is part of Number Theory Revealed: The Series. Find full
tables of contents, sample problems, hints, and appendices, as well as
updates about forthcoming related volumes here.

About the Author:

Andrew Granville is the Canada Research Chair in Number Theory at the
University of Montreal and professor of mathematics at University
College London. He has won several international writing prizes for
exposition in mathematics, including the 2008 Chauvenet Prize and the
2019 Halmos-Ford Prize, and is the author of Prime Suspects
(Princeton University Press, 2019), a beautifully illustrated graphic
novel murder mystery that explores surprising connections between the
anatomies of integers and of permutations.

#### Readership

Undergraduate and graduate students interested in introductory number theory.

#### Reviews & Endorsements

Andrew Granville has written many wonderful expository articles about number theory (especially patterns in primes), as well as dozens of research articles on many aspects of this field. It is, thus, highly appropriate that he turn all this knowledge into a series of textbooks for number theory. So if you are looking for enrichment in how the world of number theory connects together but formatted as a textbook with familiar topical arrangement, this is a great resource; I can't wait for the next two volumes to appear.

-- Karl-Dieter Crisman, MAA Reviews

#### Table of Contents

# Table of Contents

## Number Theory Revealed: A Masterclass

Table of Contents pages: 1 2 3

- 12.4. Class numbers 477477
- 12.5. Class number one 478478
- References for this chapter 481481
- Additional exercises 481481
- 12A. Composition rules: Gauss, Dirichlet, and Bhargava 485485
- 12.7. Composition and Gauss 485485
- 12.8. Dirichlet composition 488488
- 12.9. Bhargava composition 490490
- 12B. The class group 494494
- 12.10. A dictionary between binary quadratic forms and ideals 494494
- 12.11. Elements of order two in the class group 495495
- References for this chapter 496496
- 12C. Binary quadratic forms of positive discriminant 497497
- 12.12. Binary quadratic forms with positive discriminant, and continued fractions 497497
- 12.13. The set of automorphisms 499499
- 12D. Sums of three squares 500500
- 12.14. Connection between sums of 3 squares and ℎ(𝑑) 500500
- 12.15. Dirichlet’s class number formula 501501
- 12E. Sums of four squares 504504
- 12.16. Sums of four squares 504504
- 12.17. Quaternions 505505
- 12.18. The number of representations 505505
- 12F. Universality 508508
- 12.19. Universality of quadratic forms 508508
- References for this appendix 510510
- 12G. Integers represented in Apollonian circle packings 511511
- 12.20. Combining these linear transformations 511511
- Further reading on Apollonian packings 514514

- Chapter 13. The anatomy of integers 516516
- 13.1. Rough estimates for the number of integers with a fixed number of prime factors 516516
- 13.2. The number of prime factors of a typical integer 517517
- 13.3. The multiplication table problem 520520
- 13.4. Hardy and Ramanujan’s inequality 521521
- 13A. Other anatomies 522522
- 13.5. The anatomy of polynomials in finite fields 522522
- 13.6. The anatomy of permutations 523523
- More on mathematical anatomies 525525
- 13B. Dirichlet 𝐿-functions 526526
- 13.7. Dirichlet series 526526
- Further exercises 529529

- Chapter 14. Counting integral and rational points on curves, modulo 𝑝 530530
- 14.1. Diagonal quadratics 530530
- 14.2. Counting solutions to a quadratic equation and another proof of quadratic reciprocity 532532
- 14.3. Cubic equations modulo 𝑝 533533
- 14.4. The equation 𝐸_{𝑏}:𝑦²=𝑥³+𝑏 534534
- 14.5. The equation 𝑦²=𝑥³+𝑎𝑥 536536
- 14.6. A more general viewpoint on counting solutions modulo 𝑝 538538
- 14A. Gauss sums 540540
- 14.7. Identities for Gauss sums 540540
- 14.8. Dirichlet 𝐿-functions at 𝑠=1 541541
- 14.9. Jacobi sums 542542
- 14.10. The diagonal cubic, revisited 543543

- Chapter 15. Combinatorial number theory 544544
- 15.1. Partitions 544544
- 15.2. Jacobi’s triple product identity 546546
- 15.3. The Freiman-Ruzsa Theorem 548548
- 15.4. Expansion and the Plünnecke-Ruzsa inequality 551551
- 15.5. Schnirel′man’s Theorem 552552
- 15.6. Classical additive number theory 554554
- 15.7. Challenging problems 557557
- Further reading for chapter 15 558558
- 15A. Summing sets modulo 𝑝 559559
- 15.8. The Cauchy-Davenport Theorem 559559
- 15B. Summing sets of integers 561561
- 15.9. The Frobenius postage stamp problem, III 561561

- Chapter 16. The 𝑝-adic numbers 564564
- 16.1. The 𝑝-adic norm 564564
- 16.2. 𝑝-adic expansions 565565
- 16.3. 𝑝-adic roots of polynomials 566566
- 16.4. 𝑝-adic factors of a polynomial 568568
- Factoring polynomials in ℤ[𝕩] efficiently 569569
- Further reading on factoring polynomials 570570
- 16.5. Possible norms on the rationals 570570
- 16.6. Power series convergence and the 𝑝-adic logarithm 571571
- 16.7. The 𝑝-adic dilogarithm 574574
- Further reading on 𝑝-adics 575575

- Chapter 17. Rational points on elliptic curves 576576
- 17.1. The group of rational points on an elliptic curve 577577
- 17.2. Congruent number curves 580580
- 17.3. No non-trivial rational points by descent 582582
- 17.4. The group of rational points of 𝑦²=𝑥³-𝑥 582582
- 17.5. Mordell’s Theorem: 𝐸_{𝐴}(ℚ) is finitely generated 583583
- Much of the discussion in this chapter is developed from 586586
- 17.6. Some nice examples 587587
- Further reading on the basics of elliptic curves 589589
- 17A. General Mordell’s Theorem 590590
- 17.7. The growth of points 590590
- 17B. Pythagorean triangles of area 6 592592
- Integer points 593593
- There are many techniques to limit integer points in 593593
- 17C. 2-parts of abelian groups 594594
- 17.8. 2-parts of abelian, arithmetic groups 594594
- 17D. Waring’s problem 595595
- 17.9. Waring’s problem 595595
- Further reading on Waring’s problem 596596

- Hints for exercises 598598
- Recommended further reading 612612
- Index 614614