biprimes

Biprime, also called semiprime in many contexts, is a natural number that is the product of exactly two prime numbers, counted with multiplicity. Equivalently, n = p q with primes p and q, not necessarily distinct. Consequently, squares of primes are biprimes (for example 4 = 2×2 and 9 = 3×3).

Common small biprimes include 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, 39, 46, 49. These numbers are not prime and do not have more than two prime factors when multiplicities are counted. A useful characterization is Ω(n) = 2, where Ω counts prime factors with multiplicity; equivalently, ω(n) ≤ 2 with Ω(n) = 2 in the biprime case (ω is the number of distinct prime factors).

Distributionally, the number of biprimes up to x, denoted B(x), is asymptotically B(x) ~ x log log x / log x as x → ∞. This places biprimes between primes and most composites in terms of density: they are more numerous than primes for large x but still relatively sparse overall.

Computationally, testing biprimality involves factoring n into primes and checking whether it equals p q for primes p, q. If n is prime or has more than two prime factors, it is not biprime. In practice, larger instances use factoring algorithms or probabilistic primality tests as part of the decision process.

Applications and usage: biprimes are a central object in cryptography, where the difficulty of factoring semiprimes underpins certain RSA-type schemes. In analytic number theory, they serve as a natural class for studying the distribution of integers with exactly two prime factors.