Course Description
Euler's theorem, public key cryptology, pseudoprimes, multiplicative functions, primitive roots, quadratic reciprocity, continued fractions, sums of two squares and Gaussian integers.
Additional Requirements for Graduate Students:
Extra problems on weekly homework.
Athena Title
NUMBER THEORY
Prerequisite
MATH 4000/6000
Semester Course Offered
Not offered on a regular basis.
Grading System
A - F (Traditional)
Course Objectives
This course will provide an introduction to arithmetical properties of the integers, the most fundamental object in mathematics. Students will learn basic computational algorithms concerning the integers, a selection of topics concerning interesting and subtle properties of the integers, and recent applications of number theory to cryptography. Students will be expected to read and write simple proofs. They will be evaluated on thebasis of graded problem sets, hour-long exams, and a cumulative final.
Topical Outline
The basic course will include: (1) Review of mathematical induction (2) The Euclidean algorithm, unique factorization of integers (3) Congruences, the Chinese remainder theorem (4) Fermat's littlle theorem, Euler's theorem, primitive roots (5) The law of quadratic reciprocity (6) Factoring algorithms and psuedo-primality tests, Publc key cryptography Depending onthe instructor's choice, it may include other topics such as: (7) Solutions to diophantine equations, Pell's equaton, representation of integers as sums of squares, perfect numbers, elliptic curves, Fermat's last theorem (8) Multiplicative functions, Mobius inversion (9) Continued fractions (10) Primes in arithemetic progressions, the prime number theorem
Syllabus