Almost prime

In number theory, a natural number is called almost prime if there exists an absolute constant K such that the number has at most K prime factors. An almost prime n is denoted by P_r if and only if the number of prime factors of n, counted according to multiplicity, is at most r. A natural number is called k-almost prime if it has exactly k prime factors, counted with multiplicity. More formally, a number n is k-almost prime if and only if Ω(n) = k, where Ω(n) is the total number of primes in the prime factorization of n:

A natural number is thus prime if and only if it is 1-almost prime, and semiprime if and only if it is 2-almost prime. The set of k-almost primes is usually denoted by P_k. The smallest k-almost prime is 2^k. The first few k-almost primes are:

The number π_k(n) of positive integers less than or equal to n with at most k prime divisors (not necessarily distinct) is asymptotic to:

a result of Landau. See also the Hardy–Ramanujan theorem.

References

Semiprime

In mathematics, a semiprime (also called biprime or 2-almost prime, or pq number) is a natural number that is the product of two (not necessarily distinct) prime numbers. The semiprimes less than 100 are 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, 39, 46, 49, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, and 95. (sequence A001358 in OEIS). Semiprimes that are not perfect squares are called discrete, or distinct, semiprimes.

By definition, semiprime numbers have no composite factors other than themselves. For example, the number 26 is semiprime and its only factors are 1, 2, 13, and 26.

Properties

The total number of prime factors Ω(n) for a semiprime n is two, by definition. A semiprime is either a square of a prime or square-free. The square of any prime number is a semiprime, so the largest known semiprime will always be the square of the largest known prime, unless the factors of the semiprime are not known. It is conceivable, but unlikely, that a way could be found to prove a larger number is a semiprime without knowing the two factors. A composite non-divisible by primes is semiprime. Various methods, such as elliptic pseudo-curves and the Goldwasser-Kilian ECPP theorem have been used to create provable, unfactored semiprimes with hundreds of digits. These are considered novelties, since their construction method might prove vulnerable to factorization, and because it is simpler to multiply two primes together.