A semiprime, also called a 2-almost prime, biprime (Conway et al. 2008), or -number, is a composite number
that is the product of two (possibly equal) primes.
The first few are 4, 6, 9, 10, 14, 15, 21, 22, ... (OEIS A001358).
The first few semiprimes whose factors are distinct (i.e., the squarefree semiprimes)
are 6, 10, 14, 15, 21, 22, 26, 33, 34, ... (OEIS A006881).
The square of any prime number is by definition a semiprime. The largest known semiprime is therefore the square of the largest known prime.
A formula for the number of semiprimes less than or equal to is given by
|
(1)
|
where is the prime
counting function and
is the
th prime (R. G. Wilson V, pers.
comm., Feb. 7, 2006; discovered independently by E. Noel and G. Panos
around Jan. 2005, pers. comm., Jun. 13, 2006).
The numbers of semiprimes less than for
, 2, ... are 3, 34, 299, 2625, 23378, 210035, ... (OEIS A066265).
For with
and
distinct, the following congruence is satisfied:
|
(2)
|
In addition, the totient function satisfies the simple identity
|
(3)
|
Generating provable semiprimes of more than 250 digits by methods other than multiplying two primes together is nontrivial. One method is factorization. From the Cunningham
project,
and
are factored semiprimes with
274 and 301 digits. Simiarly, the indices of Mersenne
numbers that give semiprimes are 4, 9, 11, 23, 37, 41, 49, 59, 67, 83, ... (OEIS
A085724). As of 2022, the largest known indices
giving semiprimes are 1427 and 1487.
In 2005, Don Reble showed how an elliptic pseudo-curve and the Goldwasser-Kilian ECPP theorem could generate a 1084-digit provable semiprime without a known factorization (Reble 2005). Vitto (2021) subsequently found a backdoor strategy, factored Reble's 1084-digit semiprime, and introduced a new method for creating a semiprime certificate that is at least as secure as for random semiprimes of same size.
Encryption algorithms such as RSA encryption rely on special large numbers that have as their factors two large primes. The following tables lists some special semiprimes that are the product of two large (distinct) primes.
| digits in | digits in | digits in | |
| 45 | 23 | 23 | |
| 49 | 21 | 28 | |
| 51 | 22 | 29 | |
| 54 | 23 | 32 | |
| 54 | 25 | 29 | |
| 55 | 25 | 31 | |
| 64 | 32 | 32 | |
| RSA-129 | 129 | 64 | 65 |
| RSA-140 | 140 | 70 | 70 |
| RSA-155 | 155 | 78 | 78 |