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 in the 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.

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 ${\displaystyle n}$ non-divisible by primes ${\displaystyle \leq {\sqrt[{3}]{n}}}$ 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.

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