semiprime

A semiprime is a natural number that is the product of exactly two prime numbers, counted with multiplicity. That is, n = p q where p and q are primes and possibly p = q. Examples include 4 = 2×2, 6 = 2×3, 9 = 3×3, 10 = 2×5, and 14 = 2×7.

Semiprimes are a subset of composite numbers. They are not the same as numbers with more than two prime factors; for instance, 12 = 2^2×3 is not semiprime. If the two primes are distinct, the number is the product of two distinct primes; if the primes are the same, the semiprime is a prime square.

In number theory, a semiprime is defined as an integer n with Omega(n) = 2, where Omega counts prime factors with multiplicity. Thus a semiprime can be written as p q with primes p and q. Distinguishing by distinct primes, p ≠ q yields a product of two distinct primes, while p = q yields the square of a prime.

The distribution of semiprimes has been studied in analytic number theory. As x grows, the number of semiprimes not exceeding x is asymptotically proportional to x log log x divided by log x.

Determination and computation: to test whether a given number is semiprime, one can factor it and check whether exactly two prime factors (counted with multiplicity) occur. Semiprimes feature in cryptography; many public-key schemes rely on the difficulty of factoring large semiprimes into their prime components.