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Lecture# Prime Numbers and Algorithms

Description

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.

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Related concepts (30)

Primality test

A primality test is an algorithm for determining whether an input number is prime. Among other fields of mathematics, it is used for cryptography. Unlike integer factorization, primality tests do not generally give prime factors, only stating whether the input number is prime or not. Factorization is thought to be a computationally difficult problem, whereas primality testing is comparatively easy (its running time is polynomial in the size of the input).

Prime number

A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4.

AKS primality test

The AKS primality test (also known as Agrawal–Kayal–Saxena primality test and cyclotomic AKS test) is a deterministic primality-proving algorithm created and published by Manindra Agrawal, Neeraj Kayal, and Nitin Saxena, computer scientists at the Indian Institute of Technology Kanpur, on August 6, 2002, in an article titled "PRIMES is in P". The algorithm was the first one which is able to determine in polynomial time, whether a given number is prime or composite and this without relying on mathematical conjectures such as the generalized Riemann hypothesis.

Computational complexity

In computer science, the computational complexity or simply complexity of an algorithm is the amount of resources required to run it. Particular focus is given to computation time (generally measured by the number of needed elementary operations) and memory storage requirements. The complexity of a problem is the complexity of the best algorithms that allow solving the problem. The study of the complexity of explicitly given algorithms is called analysis of algorithms, while the study of the complexity of problems is called computational complexity theory.

Computational complexity theory

In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved by a computer. A computation problem is solvable by mechanical application of mathematical steps, such as an algorithm. A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used.