MATH 347 -- Number Theory

MATH 347 Number Theory
Syllabus for Spring 2016

Mondays, Wednesdays and Fridays, 10:00-10:50am

Gildemeister Hall 326

Instructor: Dr. Eric Errthum Winona Email Username: eerrthum Office: 205 Gildemeister Hall Office Hours: See homepage. Or by appointment on any day.

Text: “Introduction to Number Theory” by Peter Schumer. This book is out of print, a pdf version is available in D2L. Old used versions of the text can sometimes be found through Amazon.com or other sites ranging in price from ~$12 to ~$100.

Calculator: You are allowed at most times to use a calculator, but you must show work. At times you will be prohibited from doing specific calculations on your calculator.
You are not allowed to use your cell phone, laptop, or any other device capable of electronic communication in place of a calculator.

Course Website: http://course1.winona.edu/eerrthum/math347

Prerequisite: Passing grade in Foundations of Mathematics (MATH327 or MATH210).

About This Course: A study of primes, divisibility, congruences, number-theoretic functions, Diophantine equations, and continued fractions.

Expectations: Students who complete this course with a passing grade are expected to be able to demonstrate the following skills: (i) Mastery of prerequisite material, (ii) Proficient Modular Computing, (iii) Logical Reasoning with regard to whole numbers, (iv) Mastery of the Euclidean algorithm and its applications, (v) Solving Diophantine equations, (vi) Determine the properties of arithmetic functions and rational approximations

Grading:        Quizzes (7 @ 20 points each, drop lowest, scaled) 120 points------ 20.0%
                        Oral Homework (scaled as needed)                           65 points------ 10.8%
                        Written Homework (scaled as needed)                     65 points------ 10.8%
                        Midterms (2 @ 100 points)                                      200 points------ 33.4%
                        Final                                                                         150 points------ 25.0%
                                                                                                     --------
                                                                                                         600 points total

Grades: A = 90% (540 pts), B = 80% (480 pts), C = 70% (420 pts), D = 60% (360 pts)

Homework: Homework will be completed in two ways. Oral Homework: According to the schedule below, on Oral Homework Days students will be chosen randomly to present solutions to problems from the oral homework. Students will be graded a 0, 1, or 2 out of 2 corresponding to their level of preparedness (not necessarily correctness). When presenting a solution, you should be prepared to answer questions clarifying your work. It is not acceptable to write out a whole solution, but then when asked about a particular step to say “I don’t know.” To me this indicates you copied the homework from someone else without understanding it. Written Homework: The written portion of the homework is due the period after an Oral Homework Day. This work should be written nicely or typed, stapled, and presented in order. Each problem will be graded out of 2 corresponding to its level of correctness and clarity.

Quizzes: We will have seven (~20 minute) quizzes (see schedule below). Each quiz will count for 20 points and the lowest quiz score will be dropped from your grade. There will possibly be problems on the quizzes that did not show up explicitly in the homework. For this reason you are encouraged to work more problems than what is just assigned for homework and to ensure that you understand the concepts and calculations required in each section.

Exams: There will be three in-class exams and one comprehensive final exam. Exam dates are tentative until officially announced in class. The final exam is tentatively scheduled for Wednesday, May 4, 8:00 – 10:00am. The final exam may or may not include an oral exam to be scheduled during finals week.

Late/Missed Work: Late homework or missed quizzes will result in a score of zero. There are no make-up quizzes. Make-up exams will be given at the discretion of the instructor. If you miss class, it is your responsibility to obtain notes and assignments from fellow students. If you have an unavoidable absence, please inform the instructor beforehand.

Desire2Learn: Many course materials can be found on D2L including projects, solutions to quizzes and exams, sample exams, the Discussion boards and approximate grades.

Academic Dishonesty: Any type of academic dishonesty (cheating, copying, plagiarism, using a solutions manual to do homework, etc.) will result in failure and will be reported to school authorities. If you are having trouble with an assignment, please see the instructor first.

Note: This syllabus is subject to change if deemed necessary by the instructor.

Tentative Schedule of Events – Math 347

(subject to change)

Week Beginning	Monday	Wednesday	Friday
1/11	Introductions 1.1 Preliminaries Oral HW 1.1 (pg 4): 10, 14 Written HW 1.1 (pg 4): 22, 24	1.2 Induction/Well-Ordering 1.3 Divisibility, Division Algorithm, Congruences Oral HW 1.2 (pg 10): 8, 11 (do *not* use Binet’s formula) Written HW 1.2 (pg 10): 9a, 12, 23	1.3, cont. 1.4 Combinatorics, Pigeon Hole Principle Oral HW 1.3 (pg 15): 3a, 4ab, 15 1.4 (pg 20): 8, 20ad Written HW 1.3 (pg 15): 3b, 4c, 20 1.4 (pg 20): 9
1/18	NO CLASS MLK Day	Oral Homework Day (1.1 – 1.4)	Written Homework Due (1.1 – 1.4) Quiz 1 Algorithm Analysis
1/25	2.1 GCD & Euclidean Algorithm Oral HW 2.1 (pg 28): 1, 2, 4, 18ad, 19a, 24 Written HW 2.1 (pg 28): 12, 15, 18bc, 19bd	2.2 Congruence Equations Solving Linear Congruences Linear Diophantine Equations	2.2, cont. Chinese Remainder Theorem 2.3 Prime vs. Irreducible Oral HW 2.2 (pg 36): 2, 4, 5, 7, 9a, 10, 18 Written HW 2.2 (pg 36): 6, 14, 15, 17
2/1	Oral Homework Day (2.1 – 2.2)	Written Homework Due (2.1 – 2.2) Quiz 2 X.1 Factorization in Quadratic Domains Oral HW X.1: 1, 2 Written HW X.1: 3, 4, 5	2.3 Fundamental Theorem of Arithmetic 2.4 Sieve of Eratosthenes and Primality Testing Oral HW 2.3 (pg 41): 2, 6, 14, 17b Written HW 2.3 (pg 41): 3b, 9, 13
2/8	2.5 Euler Phi Function, Euler-Fermat Theorem Oral HW 2.5 (pg 51): 1, 5, 6 Written HW 2.5 (pg 51): 8, 9, 10	Introduction to Cryptology X.2 Diffie-Helman Key Exchange Oral HW X.2: 1 Written HW X.2: 2 Note: See here for how to use Wolfram Alpha to compute exponentiation mod m.	7.7 RSA Cryptography Oral HW 7.7 (pg 196): 5, 9 Written HW 7.7 (pg 196): 1, 10, 14
2/15	Oral Homework Day (2.3, X.1, 2.5, X.2 & 7.7)	Written Homework Due (2.3, X.1, 2.5, X.2 & 7.7) Quiz 3 Exam Review	EXAM I
2/22	4.1 Primitive Roots	4.1, cont. Using Primitive Roots Oral HW 4.1 (pg 80): 3, 5a, 9, 10, 12, 20 Written HW 4.1 (pg 80): 15, 21, 26, 27	4.2 Quadratic Residues Oral HW 4.2 (pg 84): 8, 11, 12 Written HW 4.2 (pg 84): 13, 14
2/29	4.3 Legendre Symbol 4.4 Quadratic Reciprocity (Statement and Application) Oral HW 4.3 (pg 89): 2, 3, 10 4.4 (pg 96): 1, 2 Written HW 4.3 (pg 89): 4, 19a 4.4 (pg 96): 4	X.3 Solvability of Quadratic Equations X.3 Computing Square Roots for p = 3 mod 4 Square Roots for General p (from Daniel E. Otero) Oral HW X.3: 1 Written HW X.3: 2	Oral Homework Day (4.1 – 4.4, X.3)
3/7	SPRING BREAK No Class
3/14	No Class	Written Homework Due (4.1 – 4.4, X.3) Quiz 4 X.4 Computations mod n via CRT and Lifting	X.4, cont. 2.6 Hensel's Lemma Oral HW X.4: 1, 2 2.6: 1, 3 Written HW X.4: 3, 4 2.6: 9, 10
3/21	X.5 Computations base p and Changing Bases Oral HW X.5: 1, 2, 3, 4 Written HW X.5: 5, 6	Activity: p-adic numbers	Oral Homework Day (X.4 & X.5, 2.6)
3/28	Written Homework Due (X.4 & X.5, 2.6) Quiz 5 Review for Exam	EXAM II	6.1 Finite Simple Continued Fractions Oral HW 6.1 (pg 133): 1, 4, 6 Written HW 6.1 (pg 133): 13, 16
4/4	6.3 Infinite Simple Continued Fractions Oral HW 6.3 (pg 144): 1, 2, 4, 8 Written HW 6.3 (pg 144): 5, 6	6.4 Rational Approximation of Irrationals. Oral HW 6.4 (pg 152): 3, 4, 6, 8 Written HW 6.4 (pg 152): 7, 9	6.4, cont Non-simple Continued Fractions Pade Approximation “A Short Proof of the Simple Continued Fraction Expansion of e” by Henry Cohn
4/11	Oral Homework Day (6.1, 6.3 – 6.4)	Written Homework Due (6.1, 6.3 – 6.4) Quiz 6 3.1 Arithmetic Functions Oral HW 3.1 (pg 58): 1 Written HW 3.1 (pg 58): 4, 12	NO CLASS Spring Break Day
4/18	3.2 Multiplicativity	3.2, cont 3.3 Mobius Inversion Oral HW 3.2 (pg 62): 3, 4b, 5, 13 Written HW 3.2 (pg 62): 6, 12, 14, 17	3.3, cont Oral HW 3.3 (pg 66): 1, 10, 14 Written HW 3.3 (pg 66): 4, 6, 12
4/25	Perfect Numbers Activity	Oral Homework Day (3.1 – 3.3)	Written Homework Due (3.1 – 3.3) Quiz 7 Final Review

Final Exam

Wednesday, May 4

8:00 – 10:00am

Welcome to college math!

If this is your first math class taken in college, there are some important things you need to know. College math classes are run very differently from high school math classes. On the surface it may seem they are similar as you listen to the lecture and take notes, but there are significant underlying differences. Knowing these ahead of time can help you make the most of this coming semester.

#1: College math classes generally stay on the schedule in the syllabus. If there is one day allotted to the topic that is probably all of the class time that will be spent on it, even if “most” of the students “don’t get it.”

#2: It is expected that you will read the text and do the problems in order to learn the material, even if no one checks up on you. The instructor might never collect the homework, but that doesn’t mean it doesn’t affect your grade.

#3: You will sometimes be responsible for material in the textbook that is not covered in class. If there is a text reading and/or homework problems covering a concept that was not discussed in class, you are still expected to learn it. If you don’t understand it, make an appointment with your instructor for help.

#4: Some material is covered only in class, is not in the textbook, and may not have any homework problems on it. If you miss class, you may miss content that you are responsible to know. If you have an unavoidable absence, be sure to get the notes and any announcements from a classmate.

#5: There will be test questions that don’t look “just like the homework.” In college, you are expected to focus on learning the concepts, not just memorizing how to do certain types of problems. These concepts can – and will – appear in very different forms on tests and quizzes.

#6: At times you will be expected to be able to explain why a problem is done a certain way in addition to being expected to do the problem. As you work on problems in class and on homework, don’t be satisfied with getting the correct answer; ask yourself why that method is logical, and how you could explain that logic to someone else.

#7: Most importantly, you are responsible for your own learning. If you attend class faithfully, get the notes and announcements if you have an unavoidable absence, read the text, do the homework and question yourself (as in #6), and still don’t understand something, it is up to you to get the extra help you need. Visit the instructor during office hours or make a special appointment to ask questions, form a study group, etc. There are many resources and people willing and happy to help, but you need to take the initiative and seek out the help you need.

Good luck on a happy and successful semester!

Commitment to Inclusive Excellence

WSU recognizes that our individual differences can deepen our understanding of one another and the world around us, rather than divide us. In this class, people of all ethnicities, genders and gender identities, religions, ages, sexual orientations, disabilities, socioeconomic backgrounds, regions, and nationalities are strongly encouraged to share their rich array of perspectives and experiences. If you feel your differences may in some way isolate you from WSU’s community or if you have a need of any specific accommodations, please speak with the instructor early in the semester about your concerns and what we can do together to help you become an active and engaged member of our class and community.

Campus Resources

Student Support Services, Krueger Library 219, 457-5465 www.winona.edu/studentsupportservices/
Inclusion and Diversity Office, Kryzsko Commons Room 236, 457-5595 www.winona.edu/inclusion-diversity/
Access Services, Maxwell 314, 457-5878 www.winona.edu/accessservices/
Counseling and Wellness Services, Integrated Wellness Complex 222, 457-5330 www.winona.edu/counselingcenter/
GLBTA Advocate, contact Counseling and Wellness Services for name and number of the current Advocate
Advising Services, Maxwell 314, 457-5878 www.winona.edu/advising/
Tutoring Services, Krueger Library 220, 457-5680 http://www.winona.edu/tutoring/
Writing Center, Minné Hall 348, 457-5505 www.winona.edu/writingcenter/
Math Achievement Center, Tau 313, 457-5370 http://www.winona.edu/mathematics/mac/

Details about Campus Resources

Two good places to help you find resources of all kinds on campus are Student Support/Campus Services and the Inclusion and Diversity Office. Both offices are dedicated to helping students of all races, ethnicities, economic backgrounds, nationalities, and sexual orientations. They can facilitate tutoring and point you to a wide range of resources.
If you have a disability, the Access Services office can document it for your professors and facilitate accommodations. If you have a documented disability that requires accommodation, please contact Access Services as soon as possible.
College can be very stressful. Counseling offices on both campuses are here to help you with a wide range of difficulties, ranging from sexual assault, depression, and grief after the loss of a loved one to stress management, anxiety, general adjustment to college, and many others.
The GLBTA Advocate can direct people to GLBT resources on and off campus. In addition, the advocate is responsible for documenting homophobic and transphobic incidents on campus and working with the appropriate channels to get these incidents resolved.
To find out about web registration, placement tests, program requirements and support tools to help students succeed, visit the Advising Services office and website for answers to all your questions!
On the Rochester campus, the UCR Learning Center provides help with both the development and the writing of papers.
On the Winona campus, for help with understanding the concepts of a particular class or understanding the requirements of an assignment, Tutoring Services offers three types of tutoring: drop-in appointments, 1-on-1 tutoring, and group sessions. You can visit them in the Library (220) or go on-line and use TutorTrac to schedule a session.

For help specifically with writing and the development of papers, the English department has a Writing Center that is staffed by trained graduate students pursuing their Master’s degree in English. The Writing Center is located in Minné Hall 348. You can make an appointment on the sign-up sheet on the door or call 457-5505.
For help specifically with understanding math concepts and solving math problems, the Math Achievement Center (MAC) is staffed with friendly undergraduate tutors who will help you work through difficult material. The MAC is located in Tau 313 and provides free tutoring for all students in math, statistics, or math education courses. The center is open Mon-Fri, and Sunday evening.

The Standard Disclaimer applies.