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. Though this latter task can be done in much, much more efficient ways if that's the goal! So we'll sieve the numbers up to 2n by factors up to n, then check the numbers between n and 2n for semiprimes. 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. This tells the compiler which kind of array we want for this program. The concept of the prime zeta function can be adapted to semiprimes, which defines constants like. What's more, any such semiprime cannot have a factor bigger than n since then the other factor would have to be smaller than 2! It is conceivable, but unlikely, that a way could be found to prove a larger number is a semiprime without knowing the two factors.

Although the current algorithm just returns the next semiprime, it's easy to modify it to return the factorization of the next semiprime: 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. What's more, any such semiprime cannot have a factor bigger than n since then the other factor would have to be smaller than 2! Because this latter check is O 1 , we fall in the first case you proposed. By definition, semiprime numbers have no composite factors other than themselves. It is conceivable, but unlikely, that a way could be found to prove a larger number is a semiprime without knowing the two factors. For example, the number 26 is semiprime and its only factors are 1, 2, 13, and So we'll sieve the numbers up to 2n by factors up to n, then check the numbers between n and 2n for semiprimes. 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. First some library imports: Though this latter task can be done in much, much more efficient ways if that's the goal! We can add factors to such a list like this: Maybe We'll define a new type, Primality, which we'll use for storing up to two prime factors of each number. So this is a bit like a list of Integers that's at most two integers long or else a note saying that it was three integers long or longer. It uses a few helper functions that really ought to be in the standard libraries somewhere. The first is the linear search algorithm; it just walks down a list looking for an element that satisfies a predicate. It's in Haskell, which I recognize isn't the most common language, so I'll comment inline about what each bit does. A semiprime is either a square of a prime or square-free. Each call to isSemiprime costs one multiplication, so they're O 1. The concept of the prime zeta function can be adapted to semiprimes, which defines constants like. The semiprimes less than 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 Semiprimes that are not perfect squares are called discrete, or distinct, semiprimes. This tells the compiler which kind of array we want for this program. 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.

### Video about semiprime:

## Cignol - Semiprimes

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