MATH 347 Number Theory
Syllabus for Fall 2022

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 8, 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 22

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

Aug 29

Oral Homework Day (1.1 – 1.4)

Written Homework Due (1.1 – 1.4)

Quiz 1

Algorithm Analysis

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

Sep 5

No Class
Labor Day

2.2 Congruence Equations
Solving Linear Congruences

Linear Diophantine Equations

2.2, cont.
Chinese Remainder Theorem

Oral HW
2.2 (pg 36):  2, 4, 5, 7, 9a, 10, 18

Written HW
2.2 (pg 36): 6, 14, 15, 17

Sep 12

Oral Homework Day (2.1 – 2.2)

Written Homework Due (2.1 – 2.2)

Quiz 2

2.3 Prime vs. Irreducible


X.1 Factorization in Quadratic Domains

X.1, cont

2.3 Fundamental Theorem of Arithmetic
2.4 Sieve of Eratosthenes
and Primality Testing

 

Oral HW

X.1: 1, 2
2.3 (pg 41): 2, 6, 14, 17b

 

Written HW

X.1: 3, 4, 5
2.3 (pg 41): 3b, 9, 13

Sep 19

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

Sep 26

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 3

3.1 Arithmetic Functions
3.2 Multiplicativity

 

Oral HW
3.1 (pg 58): 1

Written HW
3.1 (pg 58): 4, 12

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
X.6: 1, 2

Written HW
3.3 (pg 66): 4, 6, 12
X.6: 3, 4

Oct 10

Oral Homework Day (3.1 – 3.3)

Written Homework Due (3.1 – 3.3)

Quiz 4

6.1 Finite Simple Continued Fractions

6.1, cont.

 

6.3 Infinite Simple Continued Fractions

 

Oral HW
6.1 (pg 133): 1, 4, 6

Written HW
6.1 (pg 133): 13, 16

Oct 17

6.3, cont.

 

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

Oct 24

Oral Homework Day (6.1, 6.3 – 6.4)

Written Homework Due (6.1, 6.3 – 6.4)

Quiz 5

 

Review for Exam

EXAM II

Oct 31

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

Nov 7

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

NO CLASS
Veterans Day

Nov 14

Oral Homework Day (4.1 – 4.4, X.3)

Written Homework Due (4.1 – 4.4, X.3)

Quiz 6

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

Nov 21

X.5 Computations base p
and Changing Bases

Oral HW

X.5: 1, 2, 3, 4

 

Written HW

X.5: 5, 6

NO CLASS
Thanksgiving Break

NO CLASS
Thanksgiving Break

Nov 28

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 8

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.

 

If you or a friend has been a victim of sexual assault, dating violence, domestic violence, or stalking, you can talk to a trained, confidential advocate by calling 507.457.5610.

 Campus Resources

 

The Standard Disclaimer applies.

 

© Eric Errthum, August 2022, all rights reserved.