Overview

Description

CALENDAR DESCRIPTION:

The prime numbers, unique factorization, congruences and quadratic reciprocity. Topics include the RSA public key cryptosystem and the prime number theorem. Quantitative.

COURSE DETAILS:

Topics Include:

• Numbers and Sequences, Divisibility, Prime Numbers, Dirichlet's Theorem.
• Greatest Common Divisors, The Euclidean Algorithm, Continued Fraction Expansions, Linear Diophantine Equations.
• The Fundamental Theorem of Arithmetic.
• Introduction to Congruences, Linear Congruences, The Chinese Remainder Theorem, Solving Polynomial Congruences.
• Systems of Linear Congruences, The Distribution of Primes.
• Wilson's Theorem and Fermat's Little Theorem, Euler's Theorem, Pseudoprimes.
• The Euler Phi-Function, The Sum and Number of Divisors, Moebius Inversion, Perfect Numbers and Mersenne Primes.
• Integer factorization methods, primality testing, RSA public key crypto system.
• The Order of an Integer and Primitive Roots, Primitive Roots for Primes, Index Arithmetic, The Existence of Primitive Roots.
• Quadratic Residues and Nonresidues, The Law of Quadratic Reciprocity, The Jacobi Symbol.
• Sums of Squares, Pythagorean Triples, Infinite Descent, Pell's Equation.
• Additional topics and applications.

Grading

Materials

REQUIRED READING: