Biprimality

Biprimality refers to a property of numbers that are the product of exactly two prime numbers. A number n is biprime if it can be expressed as n = p q, where p and q are prime numbers. The primes p and q may be the same. For example, 6 is biprime because it is 2 3, and both 2 and 3 are prime. Similarly, 9 is biprime because it is 3 3, and 3 is a prime number. Numbers that are the product of two distinct primes are sometimes called distinct biprimes.

The sequence of biprime numbers starts with 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, ... These numbers are also known as semiprimes or pq-numbers. The fundamental theorem of arithmetic states that every integer greater than 1 is either a prime number itself or can be represented as the product of prime numbers, and that, moreover, this representation is unique, up to the order of the factors. Biprimes are the simplest composite numbers in this factorization scheme, being composed of exactly two prime factors.

The study of biprimes is relevant in various areas of number theory and cryptography. For instance, the security of some cryptographic algorithms, like the RSA algorithm, relies on the difficulty of factoring large numbers into their prime components, where these large numbers are often biprimes.