The goal of this course is to provide a modern treatment of number theory. The student learns more about integers and their properties, important number-theoretical ideas and their applications. The course emphasizes reading and writing proofs. Prerequisites: Grade of C or higher in both MATH 222 and MATH 225.
Prerequisite(s) / Corequisite(s):
Grade of C or higher in both MATH 222 and MATH 225.
Course Rotation for Day Program:
Offered even Fall.
Most current editions of the following:
A Friendly Introduction to Number Theory
By Silverman, Joseph (Pearson) Recommended
Elementary Number Theory
By Rosen, Kenneth R. (Addison-Wesley) Recommended
Course Learning Outcomes
Apply number theory from a procedural/computational understanding to a broad understanding encompassing logical reasoning, generalization, abstraction, and formal proof.
Describe number theory in a variety of settings, both written and orally.
Apply the basic properties of integers such as divisibility, primes, and congruence.
Identify Pythagorean triples and their properties.
Assess the value of modular expressions using properties of modular arithmetic.
Major Topics/Skills to be Covered:
Know the basic properties of integers such as divisibility, prime and congruence.
Apply the (extended) Euclidean algorithm (both iterative and recursive versions).
Apply the algorithms for modular arithmetic.
Solve linear Diophantine equations and linear congruencies.
Solve a system of linear congruencies by means of the Chinese Remainder Theorem.
Apply three famous results in congruencies: Wilson’s theorem, Fermat’s little theorem and Euler’s theorem.
Determine the number of divisors of a positive integer (that is, the arithmetic function τ) and find their sum (that is, the arithmetic function σ).
Determine the number of integers between 1 and n that are relatively prime to n, that is the arithmetic function Φ.
State and apply properties of the three functions τ, σ and Φ.
State the meaning and significance of multiplicative functions.
Apply results concerning primitive roots.
Apply the strong pseudo prime test.
Determine whether an integer is a quadratic residue.
Recommended maximum class size for this course: 20
NOTE: The intention of this master course syllabus is to provide an outline of the contents of this course, as specified by
the faculty of Columbia College, regardless of who teaches the course, when it is taught, or where it is taught. Faculty members teaching this
course for Columbia College are expected to facilitate learning pursuant to the course learning outcomes and cover the subjects listed in the Major Topics/Skills to be Covered section.
However, instructors are also encouraged to cover additional topics of interest so long as those topics are relevant to the course's
subject. The master syllabus is, therefore, prescriptive in nature but also allows for a diversity of individual approaches to course material.