### 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
• 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.

### 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.

Contact: Zeev Rudnick, rudnick@tauex.tau.ac.il, Office : Schreiber 308

Teaching assistants: Eden Kupervwasser kuperwasser@tauex.tau.ac.il, Roei Raveh roeiraveh@mail.tau.ac.il

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