Prime Numbers and Algorithms

This lecture explores the concept of prime numbers and the algorithms used to determine if a large number is prime. The instructor discusses the importance of stopping at the square root of a number when testing for primality, as well as the complexity of different algorithms. The lecture delves into modular arithmetic, demonstrating how it can simplify calculations, especially in exponentiation. By decomposing exponents into binary form, one can efficiently compute large exponentiations. The instructor also introduces probabilistic algorithms, which provide mostly correct answers by using randomization. The lecture concludes with a discussion on the computational complexity of exponentiation and the significance of efficient algorithms in the context of prime number research.