Lecture 1 (Th 6.10.): Overview
Lecture 2 (Fr 7.10.): Properties of the integers; primes and divisibility.
Lecture 3 (Th 13.10.): Greatest common divisor, division algorithm, Euclidean algorithm.
Lecture 4 (Fr 14.10.) Fundamental Theorem of Arithmetic. Are there infinitely many primes?
Lecture 5 (Th 20.10.) Euler product. Primes in arithmetic progressions.
Lecture 6 (Fr 21.10.) The Prime Number Theorem and primes in arithmetic progression.
Lecture 7 (Th 27.10.) Prime Number Theorem, Riemann Hypothesis,
Lecture 8 (Fr 28.10.) Congruences, Arithmetic mod n, Towards Fermat's Little Theorem.
Lecture 9 (Th 3.11.) Fermat's Little Theorem, linear congruences, Euler-Fermat, Euler phi function.
Lecture 10 (Fr 4.11.) Wilson's Theorem, primality tests.
Lecture 11 (Th 10.11.) Chinese Remainder Theorem.
Lecture 12 (Fr 11.11.) Chinese Remainder Theorem, Applications, Modular exponentiation.
Lecture 13 (Th 17.11.) Diffie-Hellman key exchange, computing k-th roots modulo m, trapdoor function.
Lecture 14 (Fr 18.11.) Computing k-th roots modulo m (algorithm), RSA.
Lecture 15 (Th 24.11.) Attacking/cracking RSA. Factorization methods.
Lecture 16 (Fr 25.11.) Factoring with high probability. Towards primitive roots modulo a prime.
Lecture 17 (Th 1.12.) Primitive roots modulo a prime. Index calculus.
Lecture 18 (Fr 2.12.) Index calculus. Quadratic residues.
Lecture 19 (Th 8.12.) Legendre symbol, Euler's criterion.
Lecture 20 (Fr 9.12.) Quadratic reciprocity law.
Hand in on 27th October:  Sheet 3: Q1), Q2b), Q3a), Q5c).
Hand in on 10th November:  Sheet 4: Q2b), Q3(ii), Q4c), Q6).
Hand in on 24th November:  Sheet 6: Q1a), Q2a), Q3a), Q5a).
Hand in on 8th December:  from Sheet 7: [correction:] Q4), and from Sheet 8: Q1), Q2).
(Almost) every week a new sheet will be added, and eventually solutions will be given to most of the problems.
Hint for Sheet 1, Q8b): try to give a good bound on the gcd's between any two of those numbers; which primes can divide those gcd's?
One of the most eminent number theorists in the last century was Andre Weil who has written one very nice introductory text
Furthermore, here you can find a large library of links (some are already outdated, alas) to online notes (some of them in French, German, Dutch or even Greek!), the more elementary of which cover quite similar number theoretic ground as the lectures. Not each source deals with cryptographic topics, though.