MATH 347 -- Number Theory

MATH 347 Number Theory
Syllabus for Spring 2020

Semi-Asynchronous Online

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: 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------- 21.8%
                        Oral Homework (scaled as needed)                           32 points-------- 5.8%
                        Written Homework (scaled as needed)                      32 points-------- 5.8%
                        D2L Homework (scaled as needed)                          66 points------- 12.0%
                        Midterms (3 @ 100 points)                                     300 points------- 54.5%
                                                                                                     --------
                                                                                                         550 points total

Grades: A = 90% (495 pts), B = 80% (440 pts), C = 70% (385 pts), D = 60% (330 pts)

D2L Homework: During the online portion of the course, all homework (previously oral and previously written) will be submitted online through D2L. By the due date for a homework assignment, students will be asked to complete a survey of which problems they had the most trouble with. The top 3 or 4 will have solutions posted by the instructor. The remaining questions will be randomly assigned to students in pairs: one poster, one checker. The Poster will post their solution (with necessary work) on the D2L discussion board. The Checker is responsible for checking the posted solution, asking clarification questions of the Poster, suggesting edits, etc. The whole class is encouraged to also participate in this process for each problem. A grade of 0, 1, or 2 will be given to both Poster and Checker, depending on how well they do.

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 exams. Exam dates are tentative until officially announced in class. The final exam (Exam 3) is tentatively scheduled for Wednesday, May 6, 8:00 – 10:00am.

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

1/13

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

1/20

NO CLASS

MLK Day

Oral Homework Day (1.1 – 1.4)

Written Homework Due (1.1 – 1.4)

Quiz 1

Algorithm Analysis

1/27

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

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

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

2/3

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

2/10

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

2/17

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

2/24

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

3/2

Perfect Numbers Activity

Oral Homework Day (3.1 – 3.3)

Written Homework Due (3.1 – 3.3)

Quiz 4

6.1 Finite Simple Continued Fractions

3/9

SPRING BREAK
No Class

3/16

COVID CLOSURE

3/23

COVID CLOSURE

3/30

6.1, cont.

6.3 Infinite Simple Continued Fractions

HW
6.1 (pg 133): 1, 4, 6
6.1 (pg 133): 13, 16

6.3, cont.

HW
6.3 (pg 144): 1, 2, 4, 8
6.3 (pg 144): 5, 6

6.4 Rational Approximation of Irrationals.

HW
6.4 (pg 152): 3, 4, 6, 8
6.4 (pg 152): 7, 9

Homework Survey For 6.1, 6.3 – 6.4 Due by Noon

Problems will be assigned shortly after

Optional

Non-simple Continued Fractions

Pade Approximation
“A Short Proof of the Simple
Continued Fraction Expansion of e” by Henry Cohn

4/6

D2L Homework:

Posters original post due SUNDAY at noon

Checker (and others) offer corrections/suggestions so that problem is correct by MONDAY at noon

Quiz 5

Available at 10:00am, due by 11:00am

Review for Exam

EXAM II

Available at 9:30am, due by 11:30am

4/13

4.1 Primitive Roots

HW
4.1 (pg 80): 3, 5a, 9, 10, 12, 20
4.1 (pg 80): 15, 21, 26, 27

4.2 Quadratic Residues

HW
4.2 (pg 84): 8, 11, 12

4.2 (pg 84): 13, 14

4.3 Legendre Symbol

4.4 Quadratic Reciprocity (Statement and Application)

4.3 (pg 89): 2, 3, 10
4.3 (pg 89): 4, 19a
4.4 (pg 96): 1, 2
4.4 (pg 96): 4

4/20

X.3 Solvability of Quadratic Equations
X.3 Computing Square Roots for p = 3 mod 4

X.3: 1
X.3: 2

Homework Survey For 4.1 – 4.3, X3 Due by Noon

Problems will be assigned shortly after

Optional Reading

Square Roots for General p

(from Daniel E. Otero)

D2L Homework:

Posters original post due TUESDAY at noon

Checker (and others) offer corrections/suggestions so that problem is correct by WEDNESDAY at noon

Quiz 6

Available at 10:00am, due by 11:00am

X.4 Computations mod n via
CRT and Lifting

4/27

X.4, cont.
2.6 Hensel's Lemma

HW
X.4: 1, 2
X.4: 3, 4
2.6: 1, 3
2.6: 9, 10

X.5 Computations base p
and Changing Bases

X.5: 1, 2, 3, 4

X.5: 5, 6

Homework Survey for X4 and X5 Due by Noon

Problems will be assigned shortly after

D2L Homework:

Posters original post due THURSDAY at noon

Checker (and others) offer corrections/suggestions so that problem is correct by FRIDAY at noon

Quiz 7

Available at 3:00pm, due by 4:00pm

Exam III

Wednesday, May 6

8:00 – 10:00am

The Standard Disclaimer applies.