Syllabus:

• 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 Little Theorem
• The Euclidean algorithm for polynomials
• Primitive roots
• Quadratic congruences, Legendre's symbol and quadratic reciprocity, Jacobi's symbol
• Roots of polynomial congruences, Hensel's lemma
• The Prime Number Theorem (without proof) and its applications
• Primality testing
• Public Key Cryptography
• Pell's equation
• Arithmetic in the ring of Gaussian integers, sums of two squares, Euclidean rings
• Pythagorean triples and Fermat's Last Theorem.

Bibliography

Any introductory book on number theory will be useful. For example, see:
• Elementary Number Theory, by D. Burton (available in Hebrew, published by the Open University).
• A more advanced text is "A Classical Introduction to Modern Number Theory" by Ireland and Rosen.

Course homepage: http://www.tau.ac.il/~borovoi/courses/NumberTheory/NT.html