*[*

Approved by Faculty Senate

**University Studies Course Approval Proposal
**

The Department of Mathematics and Statistics proposes the following course for inclusion in University Studies, Mathematics/Statistics Flag at Winona State University. This was approved by the full department on Thursday, February 1, 2001.

Course: Number Theory (MATH 310), 3 s.h.

Catalog Description: A study of the properties of integers. Divisibility, primes, Diophantine equations, congruences, Chinese Remainder Theorem, Euler’s phi function, Pythagorean Triples, and Fermat’s Last Theorem are some of the topics examined in the course. Property investigation and problem solving is enhanced by computer. Prerequisite: MATH 165 and MATH 210.

This is an existing course, previously approved by A2C2.

Department Contact Person for this course:

Jeffrey R. Anderson, Mathematics and Statistics Department Chair

Email janderson@winona.edu

General Discussion of University Studies – Mathematics/Statistics Flag in relation to MATH 310:

University Studies: Mathematics/Statistics Flag

The purpose of the Mathematics/Statistics Flag course requirement is to reinforce the outcomes specified for the basic skills area of mathematics/statistics. These courses are intended to provide students with significant practice in applying prerequisite mathematical or statistical knowledge.

Courses can merit the Mathematics/Statistics Flag if students will be required to make essential use throughout the semester of mathematical or statistical models appropriate to their prerequisite knowledge of those areas, and if the correct use of techniques based on such models will comprise a significant portion of a student's final grade. It is understood that mere rote computations, algebraic manipulations, or graphical design without inferential content would not merit a Mathematics/Statistics Flag.

- practice the correct application of mathematical or statistical models that are appropriate to their prerequisite knowledge of those areas;
- make proper use of modern mathematical or statistical methods appropriate to their level of prerequisite knowledge, to include, if statistics is used in a substantive way, the use of a

In MATH 310, students consider problems involving the properties of integers or subsets of the sets of integers. Frequently, it is necessary for students to conjecture what other properties may be derived from other a priori established properties. The validity of a conjecture on a few does not constitute a proof. In the end, the students' main task is to determine the validity of a conjecture for all cases through symbolic arguments and proof techniques such as through induction before it can be accepted as fact.

statistical package with graphics capability when appropriate.

An example of a problem in the course is the question of cryptograms and ciphers. What are some examples of systems of encoding messages? Which system is tougher to break than others? How can one construct an effective cipher system? Students find out that cipher systems lend themselves or not to break-ins by hackers based on the properties of integers and integer equations. Part of the exercise is for students to try to break encoded messages by testing the most commonly used ciphers. In the process of decoding messages (as in WWII), it will be necessary for students to enlist the aid of computers and software such as Mathematica.

Number Theory (MATH 310) 3 s.h.

Course Syllabus/Outline

Course Title: Number Theory MATH 310

Number of Credits: 3 S.H. Frequency of Offering: Every Spring

Prerequisite(s): MATH 165 and MATH 210

Grading: Grade only for all majors, minors, options, concentrations and licensures within the Department of Mathematics and Statistics. The P/NC option is available to others.

Course Description: A study of the properties of integers. Divisibility, primes, Diophantine equations, congruences, Chinese Remainder Theorem, Euler's phi function, Pythagorean Triples, and Fermat's Last Theorem are some of the topics examined in the course. Property investigation and problem solving is enhanced by computer.

Statement of major focus and objectives of the course: The major focus of this course is to provide students with the ability to make conjectures as to what properties are common among classes of integers and determine the validity of hypotheses and conjectures through the application of the students' knowledge of proofs. The students may have to use computer software in searching for, identifying and studying patterns that may be found among integers.

Note that a focus of the course will be to prepare students
to develop the competencies outlined in the following Minnesota Standards of
Effective Teaching Practice for Beginning Teachers: __Standard 1 -- Subject
Matter Objectives__

To develop within the future teacher ...

University Studies: Mathematics/Statistics Flag

The purpose of the Mathematics/Statistics Flag course requirement is to reinforce the outcomes specified for the basic skills area of mathematics/statistics. These courses are intended to provide students with significant practice in applying prerequisite mathematical or statistical knowledge.

Courses can merit the Mathematics/Statistics Flag if students will be required to make essential use throughout the semester of mathematical or statistical models appropriate to their prerequisite knowledge of those areas, and if the correct use of techniques based on such models will comprise a significant portion of a studentís final grade. It is understood that mere rote computations, algebraic manipulations, or graphical design without inferential content would not merit a Mathematics/Statistics Flag.

These courses must include requirements and learning activities that promote students' abilities to...

a. practice the correct application of mathematical or statistical models that are appropriate to

their prerequisite knowledge of those areas; and

b. make proper use of modern mathematical or statistical methods appropriate to their level of

prerequisite knowledge, to include, if statistics is used in a substantive way, the use of a

statistical package with graphics capability when appropriate.

Course Outline of the Major Topics and Subtopics:

- The Integers: Basic Properties; Mathematical Induction; Divisibility; Prime Numbers
- Greatest Common Divisors and Prime Factorization: The Euclidean Algorithm; The Fundamental Theorem of Arithmetic; Linear Diophantine Equations
- Congruences: Linear Congruences; The Chinese Remainder Theorem; Systems of Linear Congruences; Divisibility Tests; Wilson's Theorem; Fermat's Little Theorem; Euler's Theorem; Euler Phi Function
- Cryptology: Properties of Elemetary Cipher Systems; Character Ciphers; Block Ciphers; Exponentiation Ciphers; Public-Key Ciphers
- Pythagorean Triples: Fermat's Last Theorem

Method of Instruction: Lecture, Discussion, Group work

Evaluation Procedure:__ __Hour exams, Quizzes, Final exam,
Group work

Textbooks or Alternatives:

__Elementary Number Theory and its Applications__, Rosen

List of References and Bibliography:

__A First Course in Number Theory__: Edgar

__Elementary Number Theory__: Burton

__An Introduction to the Theory of Numbers__: Nevin & Zuckerman

__Elementary Number Theory__: Armendariz & McAdam

__Introduction to Number Theory__: Adams and Goldstein

Approval/Disapproval Recommendations

Department Recommendation: Approved Disapproved Date

Chairperson Signature Date

Dean's Recommendation: Approved Disapproved Date

Dean's Signature Date

*In the case of a Dean's recommendation to disapprove a proposal, a written rationale for the recommendation

to disapprove shall be provided to USS.

USS Recommendation: Approved Disapproved Date

University Studies Director's Signature Date

A2C2 Recommendation: Approved Disapproved Date

A2C2 Chairperson Signature Date

Faculty Senate Recommendation: Approved Disapproved Date

FA President's Signature Date

Academic VP's Recommendation: Approved Disapproved Date

VP's Signature Date

President's Decision: Approved Disapproved Date

President's Signature Date