Syllabus

Prerequisite: Maths 215 or permission of the department chairperson.

Course Description: Topics include the division algorithm; positional notation; divisibility; primes; congruences; divisibility criteria; the sigma, divisor, and phi functions; diophantine equations; linear, polynomial, and simultaneous congruences; theorems of Fermat, Euler, Lagrange, and Wilson; quadratic reciprocity.

Course Objectives: This course introduces students to a study of the deeper properties of the integers (whole numbers). Students will become familiar with various aspects of elementary number theory. ("Elementary" indicates that no complex analysis will be used; complex analysis is however important in more advanced work in number theory.) Students will become more adept at solving problems and more comfortable with proofs in mathematics. Students will appreciate number theory as one of the oldest and most fundamental products of human inquiry, and as a subject that has engaged the interest of great mathematicians throughout the centuries.

Course Rationale: Sharing its origin with mystical speculations on numbers, number theory has grown into a vast and beautiful branch of mathematics with connections with many other other branches of mathematics, as well as computer science and cryptography. It gives students exposure to problem-solving and proof in a mathematical area with minimal prerequisites. It is well suited for prospective secondary school teachers and others with an interest in pure mathematics. Maths 416 primarily serves Mathematics majors and Mathematics Teaching majors.

Course Content: topics chosen from the following list, or at the instructor’s discretion:

1. Divisibility: properties, the Euclidean algorithm, primes, Fundamental Theorem of Arithmetic, Mersenne and Fermat numbers, linear diophantine equations, least common multiples.
2. Congruences: basic properties, divisibility criteria, linear congruences, Chinese Remainder Theorem, polynomial congruences of higher degree, Lagrange’s Theorem for a prime modulus, Euler phi-function, Theorems of Fermat and Euler-Fermat, Wilson’s Theorem, pseudoprimes and Carmichael numbers, primitive roots, index arithmetic, applications to public key cryptography.
3. Quadratic residues and reciprocity: quadratic residues, Euler’s criterion, Legendre and Jacobi symbols, the law of quadratic reciprocity.
4. Multiplicative arithmetic functions: the sum and the number of divisors of an integer, formula for Euler’s phi-function, perfect numbers.
5. Nonlinear diophantine equations: Pythagorean triples, Fermat’s Last Theorem, Fermat’s Method of Infinite Descent.
6. Representations of integers as sums of powers: sum of two squares, sum of four squares, Waring’s problem.

Course Format: lecture/discussion.

Methods of Evaluating Student Performance: Possible components are homework assignments, projects, quizzes, and take-home and in-class exams. The evaluation and weight of these components are at the discretion of the instructor. Students may be evaluated on their understanding of the content material of the course, their ability to solve problems related to the course material, and on their ability to communicate mathematically to others orally and/or in writing.

Evaluation of the Course: The instruction of the course is evaluated by departmental student evaluations and peer evaluation. The course is reviewed and revised periodically by the Departmental Undergraduate Programs Committee.

K. Jones, P. Joshi, R. Bremigan, 1/2001

V. Mascioni, 10/2005