Download MATH 3527 Number Theory Spring 2016, CRN 35434

MATH 3527 Number Theory Spring 2016, CRN 35434
COURSE POLICIES & SYLLABUS
Instructor: Donald King (Associate Prof.)
Office: 447 LA
Phone: ext. 5679
E-mail: [email protected]
Lectures: Mon., Wed., Thurs. 1:35–2:40 KA 011
Office Hours: Mon. 3:00 PM–4:30 PM, Wed. 11:45 PM–1:15 PM and by appointment
Text: A Friendly Introduction to Number Theory, third edition by J. Silverman, Prentice Hall, 2006. The
author writes in a very colloquial and engaging way. At the same time many of the exercises in the book are
quite challenging. Generally there is no indication of the level of challenge of these exercises nor are there hints.
The author appeals to your desire to discover the many beautiful truths of number theory. I hope that you will
enjoy the text as much as I do.
Other materials: A calculator such as the TI-84 is essential. You should bring your calculator to class each
day. Without the calculator you won’t be able to participate in discussion and exploration.
General course objectives: The goal of this course is to introduce students to some of the basic problems and
techniques of elementary number theory. Topics include linear diophantine equations, congruences, Fermat’s
little theorem, Euler’s formula, Euler’s phi function, computing powers and roots in modular arithmetic, the
RSA encripton system, primitive roots and indices, and the law of quadratic reciprocity. As time permits, we
may cover one or more of the following topics: diophantine approximation and Pell’s equation, elliptic curves,
points on elliptic curves and Fermat’s last theorem.
Prerequisites: MATH 1342 (Calculus 2 for Engineering and Science) or the equivalent. This prerequisite is
mainly an indicator of prior experience with and mastery of a certain level of mathematical precision, logic and
sophistication needed to grasp elementary number theory. MATH 3527 will also assume a thorough mastery
of high school algebra(e.g., factoring, completing the square, quadratic formula, complex numbers). Students
should also bring to the course a strong desire and willingness to explore number facts and relationships on the
calculator and computer. Be prepared to do a lot of exact arithmetic calculations with integers in class and in
the homework. Familiarity with setting up calculations in a spreadsheet program like Microsoft Excel is also
assumed.
Other areas of emphasis: Proofs will be very important in this course (much more than in courses such
as calculus and differential equations) as in other upper level math courses like linear algebra. You will be
expected to give simple proofs on tests such as ones based on mathematical induction.
COURSE POLICIES:
1. You are expected in class each day. If for some reason, you are unable to come to a class, then (if possible)
please call or send an e-mail to let me know. Three or more unexplained absences will lower your final
grade.
2. A list of homework exercises from the textbook is given in the schedule below. (This list is subject to
revision.). Homework exercises should be done by the next class after the one in which they are assigned.
Come to class prepared to discuss these problems. Some of the exercises are especially challenging and I
have starred these exercises. You are responsible for knowing the solutions of all the unstarred homework
exercises. Please read and think about the starred exercises. I will go over the solutions of key homework
exercises in class, and/or write up and post their solutions on the Blackboard site. The questions on
quizzes will be based on the homework exercises from the textbook and the material in lectures.
3. There will be a frequent quizzes to keep students up to date on the material. Quizzes are generally 20–30
minutes long. Computers and calculators may be referenced during tests only with my explicit permission.
Cell phones must be turned off during the class, especially during exams. Makeup tests are available only
in very special circumstances (e.g., participation in university sanctioned activities such as sports, jury
duty). You must consult with me on an individual basis to convince me that you deserve a makeup test.
4. Grading:
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(a) Your grade in the course will be determined as follows:
Quizzes and tests: 60%, and Final Exam: 40%
Each quiz counts the same amount and the lowest two quiz grades will be dropped.
(b) I may decide to combine 2 or 3 quizzes into an hour exam later in the semester.
(c) I may decide to combine some assignments into a take home quiz.
(d) All students without legitimate conflicts approved by the instructor will take the final exam at the
scheduled time: April , 2016 at
. The final exam is cumulative. DO NOT MAKE TRAVEL
PLANS THAT CONFLICT WITH THE FINAL EXAM.
5. Northeastern University is committed to the principles of intellectual honesty and integrity. All members of the Northeastern community are expected to maintain complete honesty in all academic work,
presenting only that which is their own work in tests and assignments. Cheating will not be tolerated.
All incidents of cheating will be reported to the Office of Judicial Affairs. The University’s policy on
cheating and related disciplinary actions are detailed in the Student Handbook and at the following web
site http://www.northeastern.edu/osccr/academicintegrity.
6. It is the student’s responsibility to be aware of what happens in the classroom, including announcements
of possible exam (or quiz) date changes, material that will be covered and changes to the syllabus, which
may occur.
7. If you have issues with this course and/or instructor which you are not comfortable discussing with me or
cant resolve to your satisfaction, you should contact the teaching director of the mathematics department,
Prof. David Massey, [email protected].
8. The letter grade will be determined from the numerical average using the scale:
A: 93-100; A-:90-92; B+:87-89; B: 83-86; B-: 80-82; C+:77-79; C:73-76; C-:70-72; D+: 66-69; D:58-65;
D-:50-57; F:0-49; U: 0-72.
Borderline cases will be determined by the final exam score. Thus, a student whose course average is 89,
but scores 90 or above on the final exam will receive an A-. Whereas a student whose course average is
89.5 but whose final exam score is 88 will receive a B+.
As a matter of Math Department policy: The I grade (incomplete) will be given only rarely. It is intended
to cover real emergency situations in which a student who is doing reasonably well (C- or better) is unable,
due to circumstances beyond the student’s control, to complete all course requirements (e.g., is unable to
take the final exam due to hospitalization). An I may not be used to rescue a failing grade, or to postpone
the final.
9. Here are some important dates. The last day to drop a course without receiving a ’W’ grade is February
1. The last day to file a conflict final form is Wednesday, February 3. The last day of regular fall classes
is Wednesday, April 20. The last date to drop a class with a ’W’ grade is April 21 (Reading Day). Please
keep April 21 (Reading Day) open in your schedule. There will probably be a review session on that date.
10. The university will be closed (i.e., no classes) on the following days:
Monday, January 18, MLK Birthday
Monday, February 22, Presidents Day
Monday, March 7 Friday, March 11, Spring Break
Monday, April 18, Patriots Day
11. If you want to see me, but cannot do so during my office hours, then please see me before or after any
class to set up a convenient time.
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Tentative Schedule for MATH 3527 — Spring 2016
Day
Mon
Wed
Thur
Mon
Wed
Thur
Date
1/11
1/13
1/14
1/18
1/20
1/21
Mon
Wed
Thur
1/25
1/27
1/28
Mon
Wed
Thur
2/1
2/3
2/4
Mon
Wed
Thur
2/8
2/10
2/11
Mon
Wed
Thur
Mon
Wed
Thur
2/15
2/17
2/18
2/22
2/24
2/25
Mon
Wed
Thur
2/29
3/2
3/3
Mon
Wed
Thur
3/14
3/16
3/17
Mon
Wed
Thur
3/21
3/23
3/24
Mon
Wed
Thur
3/28
3/30
3/31
Mon
4/4
Wed
Thur
Mon
4/6
4/7
4/11
Section
Ch. 1 - What is Number Theory?
Ch.2 - Pythagorean Triples
Ch.3 - Pythagorean Triples and the Unit Circle
MLK Birthday – No Classes
Ch.4 - Sums of Higher Powers and Fermat’s Last Theorem
Ch.5 - Divisibility and the Greatest Common Divisor
Quiz 1
Ch.6 - Linear Equations and the Greatest Common Divisor
Ch.7 - Factorization and the Fundamental Theorem of Arithmetic
Ch.7 - Factorization and the Fundamental Theorem of Arithmetic
Quiz 2
Ch.8 - Congruences
Ch.9 - Congruences, Powers, and Fermat’s Little Theorem
Ch.10 - Congruences, Powers, and Euler’s Formula
Quiz 3
Ch.11 - Euler’s Phi Function and the Chinese Remainder Theorem
Ch.12 - Prime Numbers
Ch. 12 Prime Numbers
Quiz 4
Ch.13 - Counting primes
Ch.14 - Mersenne Primes
Ch.15 - Mersenne Primes and Perfect Numbers
Presidents Day – No Classes
Ch.16 - Powers Modulo m and Successive Squaring
Ch.16 - Powers Modulo m and Successive Squaring
Quiz 5
Ch.17 - Computing k-th Roots Modulo m
Ch.18 - Powers, Roots, and ”Unbreakable Codes”
Ch.18 - Powers, Roots, and ”Unbreakable Codes”
Quiz 6
SPRING BREAK 3/5–3/13
Ch.19 - Primality Testing and Carmichael Numbers
Ch.20 - Euler’s Phi Function and Sums of Divisors
Ch.20 - Euler’s Phi Function and Sums of Divisors
Quiz 7
Ch.21 - Powers Modulo p and Primitive Roots
Ch.22 - Primitive Roots and Indices
Ch.23 - Squares Modulo p
Quiz 8
Ch.24 - Is (-1) a square modulo p?
Ch.24 - Is 2 a square modulo p?
Ch. 25 - Quadratic Reciprocity
Ch. 25 - Quadratic Reciprocity
Quiz 9
Ch.26 - Which Primes are Sums of Two Squares?
Ch.27 - Which Numbers are Sums of Two Squares?
Ch. 29- Square Triangular Numbers Revisited
Quiz 10
3
HW
1*, 2, 3* (read), 4*(read)
4*, 5, 6
1, 2a, 2b*(read), 3, 4*
2
1, 4
1, 2, 6ab
2
3, 4
2, 3, 4
1, 4
2, 3a, 3b*
1, 2, 5, 6
2
6
1b, 2, 3, 5a, 6a*
1, 2
1, 2, 3abc, 3d*, 3e*
1
3
1, 2, 3a, 5ab
Ch. 17: 4*
1
2*, 3, 4, 7ab
1
2a, 2b*
2, 4, 6, 7a, 8
1, 2, 4
1, 3ab
1ab
1c, 3
1
Ch. 24:1d
1, 2, 4
2, 3, 4
1, 2, 3
1, 3, 4
Day
Wed
Thur
Mon
Wed
Thur
Fri
Date
4/13
4/14
4/18
4/20
4/21
4/22-4/29
Section
Ch. 30 - Pell’s Equation
Review for Final Examination
Patriot’s Day - No Classes
Review for Final Examination
Reading Day
Final Exams
4
HW
1, 2, 4b*