Introduction to Number Theory
Prof. Zeev Rudnick
Fall 2016/17, Tel Aviv
Schedule: Monday 10-12,Shenkar 104, Wednesday 11-13 Dan David 203
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,
- The Euclidean algorithm for polynomials over a finite field
- Primitive roots
- Quadratic congruences, Legendre's symbol and quadratic reciprocity,
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,
sums of two squares and primes represented by binary quadratic forms.
- Pythagorean triples and Fermat's Last Theorem
Any introductory book on number theory will be useful. For example,
"Elementary Number Theory" by W.J. LeVeque, Dover 1990.
- Elementary Number Theory, by D. Burton (available in Hebrew,
published by the Open University).
- A friendly introduction to number theory, by Joseph H. Silverman
(third edition, Prentice Hall)
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
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.
- Assignment 1, due date: Monday November 14, 2016
- Assignment 2, due date: Monday November 21, 2016
- Assignment 3, due date: Monday November 28, 2016
Assignment 4, due date: Monday December 5, 2016
Assignment 5, due date: Monday December 12, 2016
Assignment 6, due date: Monday December 19, 2016
Assignment 7, due date: Monday December 26, 2016
Assignment 8, due date: Monday January 2, 2017
Assignment 9, due date: Monday January 9, 2017
Assignment 10, due date: Monday January 16, 2017
Assignment 11, not to be handed in.
Contact me at: email@example.com, Office : Schreiber 316, tel: 640-7806
Mailbox 085, first floor of Schreiber
Course homepage: http://www.math.tau.ac.il/~rudnick/courses/int_numth.html