Biprime

A biprime is a natural number that can be written as the product of two prime numbers. The primes may be equal, in which case the biprime is a square of a prime; for example, 4 = 2×2 and 9 = 3×3. In many contexts, a biprime is synonymous with a semiprime, i.e., a product of two primes, with the distinction that some authors require the two primes to be distinct.

Examples of biprimes include 4, 6, 9, 10, 14, 15, and 21.

All biprimes are composite numbers with exactly two prime factors when counted with multiplicity. They are not divisible by three or more primes, and their prime factorization has length two.

There are infinitely many biprimes. The number of biprimes less than or equal to x is asymptotically x log log x / log x as x grows large.

In cryptography, biprimes formed from two large primes are used as RSA moduli. Generating such numbers involves selecting large primes and multiplying them, or using probabilistic primality tests and factoring methods to verify the two-prime structure.

Related concepts include semiprimes and more generally numbers with a fixed total number of prime factors. This overview treats biprimes as the set of integers expressible as p×q with primes p and q.

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