**AMS/MAA Textbooks**

Volume: 39;
2018;
283 pp;
Hardcover

MSC: Primary 11;

Print ISBN: 978-1-4704-4348-1

Product Code: TEXT/39

List Price: $60.00

AMS Member Price: $45.00

MAA Member Price: $45.00

**Electronic ISBN: 978-1-4704-4684-0
Product Code: TEXT/39.E**

List Price: $60.00

AMS Member Price: $45.00

MAA Member Price: $45.00

#### Supplemental Materials

# An Open Door to Number Theory

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*Duff Campbell*

MAA Press: An Imprint of the American Mathematical Society

A well-written, inviting textbook designed for
a one-semester, junior-level course in elementary number theory. The
intended audience will have had exposure to proof writing, but not
necessarily to abstract algebra. That audience will be well prepared
by this text for a second-semester course focusing on algebraic number
theory. The approach throughout is geometric and intuitive; there are
over 400 carefully designed exercises, which include a balance of
calculations, conjectures, and proofs. There are also nine
substantial student projects on topics not usually covered in a
first-semester course, including Bernoulli numbers and polynomials,
geometric approaches to number theory, the \(p\)-adic numbers,
quadratic extensions of the integers, and arithmetic generating
functions.

An instructor's manual for this title is available electronically
to those instructors who have already adopted the textbook for
classroom use. Please send email to textbooks@ams.org for more
information.

#### Table of Contents

# Table of Contents

## An Open Door to Number Theory

- Cover Cover11
- Title page iii4
- 1. The Integers, \Z 114
- 1. Number systems 114
- 2. Rings and fields 316
- 3. Some fundamental facts about \Z and \N 720
- 4. Proofs by induction 1326
- 5. The binomial theorem 1831
- 6. The fundamental theorem of arithmetic (foreshadowing) 2639
- 7. Divisibility 2942
- 8. Greatest common divisors 3144
- 9. The Euclidean algorithm 3346
- 10. The amazing array 3952
- 11. Convergents 4255
- 12. The amazing super-array 4962
- 13. The modified division algorithm 5669
- 14. Why does the amazing array work? 5871
- 15. Primes 6174
- 16. The proof of the fundamental theorem of arithmetic 6477
- 17. Unique factorization in other rings 6881

- 2. Modular Arithmetic in \Z/π\Z 7184
- 18. The integers mod π, \Z/π\Z 7184
- 19. Congruences 7689
- 20. Units and zero-divisors in \Z/π\Z 8194
- 21. Cancellation law in \Z/π\Z 8598
- 22. Solving linear equations in \Z/π\Z 87100
- 23. Solving polynomial equations in \Z/π\Z 88101
- 24. Solving systems of linear equations in \Z/π\Z 95108
- 25. Lifting roots in \Z/πβΏ\Z 103116
- 26. Wilsonβs theorem and its converse 108121
- 27. Calculating π(π) 110123
- 28. Eulerβs and Fermatβs theorems 115128
- 29. The order of an integer modulo π 118131
- 30. Divisibility tests 122135

- 3. Quadratic Extensions of the Integers, \Z[βπ] 127140
- 4. An Interlude of Analytic Number Theory 153166
- 5. Quadratic Residues 157170
- 39. Perfect squares 157170
- 40. Quadratic residues 160173
- 41. Calculating the Legendre symbol (hard way) 167180
- 42. The arithmetic of \Z[β-2] and the Legendre symbol \Leg{-2}π 169182
- 43. Gaussβs lemma 171184
- 44. Calculating the Legendre symbol (easier way) 174187
- 45. The arithmetic of \Z[β-3] 180193
- 46. The arithmetic of \Z[π] 182195
- 47. Calculating the Legendre symbol (easiest way) 193206
- 48. The Jacobi symbol 197210

- 6. Further Topics 203216
- Appendix A. Tables 223236
- Appendix B. Projects 233246
- Bibliography 279292
- Index 281294
- Back Cover Back Cover1297