Pseudoprimes

Pseudoprimes are composite numbers that pass the Fermat primality test, a test used to determine whether a number is prime or not. The test involves checking whether a number n is a witness to the compositeness of n-1. In other words, if there exists a number a such that a^(n-1) ≡ 1 (mod n), then n is a pseudoprime base a.

The concept of pseudoprimes was first introduced by Adrien-Marie Legendre in the 19th century. However, it was not until the development of the Fermat primality test by Albert Girard and Pierre de Fermat in the 17th century that pseudoprimes became a significant area of study in number theory.

Pseudoprimes can be classified into different categories. The most common type is Carmichael numbers, also known as absolute pseudoprimes, which are composite numbers that pass the Fermat primality test for all bases up to a certain range. These numbers are named after Robert Carmichael, who discovered them in the early 20th century.

The study of pseudoprimes is important in number theory because it helps us understand the weak points of the Fermat primality test. While the test can efficiently identify prime numbers, it can also lead to the discovery of composite numbers that mimic the properties of prime numbers. Understanding pseudoprimes can lead to the development of more robust primality tests, which is essential for various applications in cryptography and code theory.

Pseudoprimes have been extensively studied and classified over the years. The smallest Carmichael number is 561, and the largest known pseudoprime is 2^43,090 + 1. The study of pseudoprimes continues to be an active area of research in number theory.