Introduction to Higher Mathematics
Mathematics 760 – Unit 4: Number Theory
Brown University – Spring, 2020
Professor Joseph Silverman

 Topic We will study topics from number theory. The text for this unit of Math 760 may be downloaded from: Introduction to Higher Mathematics: Unit 4: Number Theory . Mathematics Department, Kassar House, Room 202 863-1124 jhs@math.brown.edu Math 750–760 Course Website Math 750–760 Unit #4 Website Mondays 2:00-2:45pm, Thursdays 9:15-10:00am Office hours start Thursday January 23 and end on Thursday April 2 Click here for Information about Office Hours and Other Academic Resources TuTh 1:00–2:20pm (J hour) Barus & Holley 163 Homework assignments are posted below. It is best if you do the reading before the class where we cover the material.

 Class Date Chapter Reading Topic Homework HW Due Date Optional Challenge Problems** 1 Thurs, Jan 23 1 §§1.1–1.2 Intro to Number Theory # 1.1, 1.2, 1.5, 1.6, 1.7, 1.9 Tues, Jan 28 # 1.8 2 Tues, Jan 28 2 §§2.1–2.2 Pythagorean Triples # 2.1, 2.2, 2.4, 2.7, 2.8, 2.11 Thurs Jan 30 # 2.3, 2.13 3 Thurs, Jan 30 3 §§3.1–3.2 Divisibility # 3.1, 3.3, 3.4(a,b,c), 3.6, 3.7 Tues, Feb 4 # 3.2, 3.5, 3.8 4 Thurs, Feb 4 4 §§4.1–4.2 Primes 1 # 4.1, 4.2, 4.3, 4.4(a,c,d), 4.5, 4.6 Thurs, Feb 6 5 Thurs, Feb 6 5 §§5.1–5.2 Congruences # 5.1(b), 5.2, 5.3, 5.5(a), 5.6, 5.9, 5.10, 5.12 Tues, Feb 11 # 5.11 6 Tues, Feb 11 6 §§6.1–6.2 Primes 2 # 6.1, 6.2, 6.3(a,b), 6.5(a,b,c), 6.7, 6.9 Thurs, Feb 13 # 6.4, 6.5(d,e,f) 7 Thurs, Feb 13 7 §§7.1–7.2 Fibonacci Sequence # 7.1, 7.2, 7.4(a,b,d), 7.5(a,b,c), 7.10 Thurs, Feb 20 # 7.6, 7.7, 7.14 Tues, Feb 18 — Brown closed 8 Thurs, Feb 20 8 §8.1 Squares Mod p # 8.1, 8.2(a,b,c,d) Tues, Feb 25 # 8.2(e) 9 Tues, Feb 25 9 §§9.1–9.2 Quadratic Reciprocity # 9.1, 9.2, 9.3, 9.4, 9.9(a) Thurs, Feb 27 # 9.6, 9.10
** Optional Challenge Problems: For those who enjoy working on challenging problems for the sake of working on challenging problems. They will not earn you extra credit, but doing them will earn you the warm satisfaction of doing a hard problem and gaining further mathematical insight.

Course and Unit Goals: Math 750-760 is a year-long class that exposes students to six basic areas of mathematics. It is team taught by several members of the faculty. Fall topics include logic, set theory, combinatorics, and analysis. Spring topics include number theory, abstract algebra, and geometry. The class emphasizes rigorous proofs and concrete interesting examples. The specific goals for the Number Theory Unit are to learn fundamentals topics from number theory.

Learning Activities and Time Allocation: Learning activities for this part of the course include class attendance and weekly problem sets. The time to complete these activities are (1) attending lectures, approximately 3 hours/week; (2) reading material in the book, approximately 2 hours/week; (3) working on the problem sets, approximately 7 hours/week.

Assessment: Course grades will be determined by the quality of problem sets submitted and by grades on the unit exams (if any).

Expectations of Students: It is expected that students will attend all lectures and participate in class discussion in an appropriate manner. Assignments are due on the listed dates. All students are expected to abide by Brown's academic code, which may found here

Syllabus:

• Introduction to Number Theory
• Pythagorean Triples
• Divisibility and Greatest Common Divisors
• Primes
• Congruences
• The Fibonacci Sequence