Week1: What are Prime Numbers?
Introduction to basic concepts and properties of prime numbers, such as infinitude of prime numbers, counting prime numbers, and the Basel problem and its relation with the Riemann Hypothesis.
Week 2: Sums of Two Squares
Introduction to the modular arithmetic and its applications to number theory, including Fermat's Little Theorem, Wilson's Theorem, and Fermat's theorem on sums of two squares.
Week 3: The Reciprocity Laws
Introduction to the quadratic reciprocity laws proved by Gauss. Several generalizations of the quadratic reciprocity laws are also explained.
Week 4: Prime Numbers and Cryptography
Introduction to cryptography, and the construction practical cryptosystems using prime numbers. More recent topics on elliptic curve cryptosystems are also explained.
Week 5: Mystery of Prime Numbers: Past, Present, and Future
Introduction to several open problems and conjectures on prime numbers, including the Birch and Swinnerton-Dyer conjecture and the ABC conjecture.