Introduction to Number Theory

Prof. Zeev Rudnick

Fall 2011/12, Tel Aviv

Schedule: Monday 10-12,Dan David 002, Wednesday 10-12 Dach 005

Syllabus:

The course is an introductory course in basic number theory. It assumes very little background beyond linear algebra and a solid course of first year calculus. The topics include

The Euclidean algorithm, greatest common divisor, unique factorisation into primes, linear Diophantine equations
Continued fractions
Congruences, the Chinese Remainder Theorem
The multiplicative group of reduced residue classes modulo n, Fermat's theorem
The Euclidean algorithm for polynomials over a finite field
Primitive roots
Quadratic congruences, Legendre's symbol and quadratic reciprocity, Jacobi's symbol
Roots of polynomial congruences, Hensel's lemma
The Prime Number Theorem and its applications
Primality testing
Public Key Cryptography
Pell's equation
Diophantine approximation, Liouville's theorem on rational approximations to algebraic numbers, Thue's equation
Arithmetic in rings of integers of quadratic fields, Euclidean rings, sums of two squares and primes represented by binary quadratic forms.
Pythagorean triples and Fermat's Last Theorem

Bibliography

Any introductory book on number theory will be useful. For example, see:

"Elementary Number Theory" by W.J. LeVeque, Dover 1990.
Elementary Number Theory, by D. Burton (available in Hebrew, published by the Open University).
Elementary Number Theory: Primes, Congruences and Secrets, by William Stein (see online version).
A more advanced text is "A classical introduction to modern number theory" by Ireland and Rosen

Homework:

This will be an important part of the course. In order to be eligible to take the final exam, at least 50% of the assignments have to be turned in on the week of their due date. 10% of the final grade will be determined from the homework scores, which will be obtained as the average grade of a certain number of assignments.

Assignments #1 , #2 , #3 , #4 , #5 , #6 , #7 , #8 , #9 , #10 , #11 , #12 .

solution to assignment #10 .

Final Exam: will take place on February 15, 2012. Here is a sample exam

Contact me at: rudnick@post.tau.ac.il, Office : Schreiber 316, tel: 640-7806

Mailbox 073, first floor of Schreiber

Course homepage: http://www.math.tau.ac.il/~rudnick/courses/int_numth.html