Introduction to Number Theory
Prof. Zeev Rudnick
Fall 2016/17, Tel Aviv
Schedule: Monday 1012,Shenkar 104, Wednesday 1113 Dan David 203
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).
 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
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:
 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: rudnick@post.tau.ac.il, Office : Schreiber 316, tel: 6407806
Mailbox 085, first floor of Schreiber
Course homepage: http://www.math.tau.ac.il/~rudnick/courses/int_numth.html