Elementary Number Theory Course Proposal

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1. Course Description Form – Honors College
The Honors Course Proposal must include the following information:
Proposed Course Title –
Elementary Number Theory
50-word Course Description –
This course engages students in the systematic study of problems in elementary
number theory using definitions and logical deductions from these definitions.
Emphasis will be on developing critical thinking and applications to modern
problems.
Course Justification Course Goals and Objectives –
Mathematics Component Outcomes:
 Students will articulate quantitative literacy in logic, patterns, and
relationships.
 Students will explain/apply/interpret key mathematical concepts and apply
appropriate quantitative tools to everyday experience.
Core Objectives/Competencies Outcomes:
 Critical Thinking
o Students will demonstrate creative thinking, innovation, inquiry,
analysis, evaluation, and synthesis of information.
 Communication
o Students will effectively develop, interpret, and express ideas
through written, oral, and visual communication
 Empirical and Quantitative Skills
o Students will manipulate and analyze numerical data or observable
facts resulting in informed conclusions.
Additional department or instructor course outcomes (optional):
 Students will examine problems deeply and formulate careful, rigorous
mathematical proofs.
 Students will complete proofs using the foundational ideas in elementary
number theory.
 Students will apply the mathematics learned to different types of problems.
In short, the students will connect the ideas from number theory to thinking
critically in a wide variety of areas.
Brief Description of Instructional Methodologies –
 Class discussions
 Homework
 Journal writing

Small group discussions
Assessment of Student Learning –
 Number Theory Notebook
 Homework
 Journal
 Weekly Quizzes
 Midterm
 Final Exam
 Total Points:
50 points (10%)
100 points (20%)
50 points (10%)
100 points (20%)
100 points (20%)
100 points (20%)
500 points (100%)
Course Outline–
Week 1: The integers – bases.
Axioms for the integers, ordering and
divisibility
Week 9: Complete and reduced
residue systems. Fermat’s Little
Theorem and Wilson’s Theorem
Week 2: Logic
Week 10: Chinese Remainder
Theorem. Polynomials and division
algorithm
Week 3: Trichotomy, factorial notation,
perfect squares. Properties of
multiplication, positives and negatives
Week 11: Factor theorem, roots of
polynomials, and order of elements
Week 4: Modular arithmetic,
properties of congruences. Weak
induction, well-ordering
Week 12: Z mod p is cyclic. Units and
Euler’s Phi Function
Week 5: The division algorithm.
Euclid’s algorithm and the magic table
Week 13: Coding theory and raising
numbers to large powers
Week 6: Greatest common divisor,
least common multiple. Strong
induction, primes, prime factorization
Week 14: Weak, strong pseudo-prime
tests
Week 15: Electronic coin toss
Week 7: Uniqueness, relations, and
equivalence relations
Week 16: Final exam
Week 8: Review and Test 1
Suggested Textbooks and Other Learning Resources –
Students will read parts or all of the following:
Burger, Edward B., and Michael Starbird. The 5 Elements of Effective Thinking.
Princeton University Press, 2012. Print
Bibliography –
Burger, Edward B., and Michael P. Starbird. The Heart of Mathematics: An
Invitation to Effective Thinking. Springer, 2005. Print.
Courant, Richard, and Herbert Robbins. What is Mathematics?: An Elementary
Approach to Ideas and Methods. Oxford University Press, 1996. Print.
LeVeque, William Judson. Fundamentals of Number Theory. Courier Dover
Publications, 1996. Print.
Marshall, David C., Edward Odell, and Michael P. Starbird. Number Theory
through Inquiry. Mathematical Association of America, 2007. Print.
Niven, Ivan, Herbert S. Zuckerman, and Hugh L. Montgomery. An Introduction
to the Theory of Numbers. John Wiley & Sons, 2008. Print.
Rosen, Kenneth H. Elementary Number Theory and Its Applications. AddisonWesley, 1993. Print
Sierpinski, Waclaw. Elementary Theory of Numbers: Second English Edition.
Ed. A. Schinzel. Elsevier, 1988. Print.
Course Resources –
A class with whiteboards.
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