Semiprimes

A semiprime is a natural number that is the product of two primes, counting multiplicity. Equivalently, it has exactly two prime factors when multiplicities are considered, so any n = p q with primes p ≤ q qualifies. This includes the case p = q, which gives squares of primes such as 4 = 2×2 or 9 = 3×3.

Semiprimes are composite and not prime, occupying a middle ground between primes and numbers with more extensive prime factorization. In terms of structure, every semiprime can be written as the product of two primes, and no semiprime has more than two prime factors counting multiplicity.

Common examples include 4, 6, 9, 10, 14, 15, 21, 25, 26, 33, 34, 35, and 49. More generally, semiprimes are precisely the numbers with Omega(n) = 2, where Omega counts prime factors with repetition.

Counting and distribution: Let S(x) denote the number of semiprimes not exceeding x. Asymptotically, S(x) is ~ x log log x / log x. Consequently, the proportion of semiprimes among integers up to x tends to zero as x grows, even though the count grows without bound.

Applications and significance: Semiprimes are central in certain areas of number theory and cryptography. Many public-key systems, such as RSA, rely on the difficulty of factoring large semiprimes n = p q, where p and q are large primes. Generating such semiprimes involves selecting two primes and multiplying them. Semiprimes also serve as a simple, well-defined class of composites for testing factorization algorithms and primality-related methods.