MATH 347
Number Theory
Syllabus for Fall 2021
Mondays, Wednesdays
and Fridays, 11:00-11:50am
Gildemeister Hall 325
Instructor: Dr. Eric Errthum Winona Email: eerrthum [at] winona [dot] edu 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: Foundations of Mathematics (MATH327).
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 and equation solving, (iii) Logical reasoning with regard to whole numbers, (iv) Mastery of the Euclidean algorithm and its applications, (v) Solving Diophantine equations, (vi) Determining the properties of arithmetic functions
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 Thursday, December 9, 8:00 – 10:00am. The final exam may or may not include an oral exam to be scheduled during finals week.
Reclaiming Points: If you miss points on a quiz, you are allowed
to come to my office during office hours (or by appointment) and request
another try to reclaim the lost points. Note: requests will be granted or
denied by discretion of the instructor. In particular,
requests will be denied if: problems are of a memorization nature, it is
a bonus problem, or you already got 80% or better on the problem.
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: Course materials can be found on D2L/Brightspace including approximate grades. If you ever want an exact look at your grade, email the instructor directly.
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: using tests/quizzes from a previous semester to study without instructor permission is a form of cheating.
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 |
Aug 23 |
Introductions 1.1 Preliminaries
Written HW |
1.2
Induction/Well-Ordering Oral HW |
1.3, cont.
1.4 (pg 20): 8, 20ad 1.4 (pg 20): 9 |
Aug 30 |
Oral Homework Day (1.1 – 1.4) |
Written Homework Due (1.1 – 1.4) Quiz 1 Algorithm Analysis |
2.1 GCD & Euclidean Algorithm
2.1 (pg 28): 1, 2, 4, 18ad, 19a, 24 Written HW 2.1 (pg 28): 12, 15, 18bc, 19bd |
Sep 6 |
No Class |
2.2 Congruence
Equations Linear Diophantine
Equations |
2.2, cont. Oral HW |
Sep 13 |
Oral Homework Day (2.1 – 2.2) |
Written Homework Due (2.1 – 2.2) Quiz 2 2.3 Prime vs. Irreducible
|
X.1, cont 2.3 Fundamental
Theorem of Arithmetic Oral HW X.1: 1, 2 Written HW X.1: 3, 4, 5 |
Sep 20 |
2.5 Euler Phi
Function, Euler-Fermat Theorem Oral HW |
Introduction to Cryptology X.2 Diffie-Helman Key Exchange Oral HW Note: See
here for how to use Wolfram Alpha to compute exponentiation mod m. |
7.7 RSA Cryptography Oral HW |
Sep 27 |
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 |
Oct 4 |
3.1 Arithmetic
Functions Oral HW |
NO CLASS |
3.2, cont 3.3 Mobius Inversion Oral HW |
Oct 11 |
3.3, cont Oral HW |
Oral Homework Day (3.1 – 3.3) |
Written Homework Due (3.1 – 3.3) Quiz 4 |
Oct 18 |
6.1, cont. 6.3 Infinite Simple Continued Fractions Oral HW |
6.3, cont. Oral HW |
6.4 Rational Approximation of Irrationals. Oral HW |
Oct 25 |
6.4, cont Non-simple Continued Fractions Pade Approximation
|
Oral Homework Day (6.1, 6.3 – 6.4) |
Written Homework Due (6.1, 6.3 – 6.4) Quiz 5 Review for Exam |
Nov 1 |
EXAM II |
4.1 Primitive Roots |
4.1, cont. Oral HW |
Nov 8 |
4.2 Quadratic Residues
|
4.3 Legendre Symbol 4.4 Quadratic
Reciprocity (Statement and Application) Oral HW 4.3 (pg 89): 2,
3, 10 Written HW |
X.3 Solvability of Quadratic
Equations (from Daniel E. Otero) Oral HW |
Nov 15 |
Oral Homework Day (4.1 – 4.4, X.3) |
Written Homework Due (4.1 – 4.4, X.3) Quiz 6 |
X.4,
cont. Written HW |
Nov 22 |
X.5 Computations base p Oral HW Written HW |
NO CLASS |
NO CLASS |
Nov 29 |
Activity: p-adic numbers |
Oral Homework Day (X.4 & X.5, 2.6) |
Written Homework Due (X.4 & X.5, 2.6) Quiz 7 Final Review |
Exam III, Final Exam
Thursday, December 9
8:00 – 10:00am
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 for 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.
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The Standard Disclaimer
applies.
© Eric Errthum, August 2021, all rights reserved.