mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a semiprime is a

natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...

that is the

product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...

of exactly two

prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

s. The two primes in the product may equal each other, so the semiprimes include the

squares In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...

of prime numbers. Because there are infinitely many prime numbers, there are also infinitely many semiprimes. Semiprimes are also called biprimes.

Examples and variations

The semiprimes less than 100 are: Semiprimes that are not square numbers are called discrete, distinct, or squarefree semiprimes: The semiprimes are the case

k=2

of the

k

almost prime In number theory, a natural number is called ''k''-almost prime if it has ''k'' prime factors. 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 facto ...

s, numbers with exactly

k

prime factors. However some sources use "semiprime" to refer to a larger set of numbers, the numbers with at most two prime factors (including unit (1), primes, and semiprimes). These are:

Formula for number of semiprimes

A semiprime counting formula was discovered by E. Noel and G. Panos in 2005. Let

\pi_2(n)

denote the number of semiprimes less than or equal to n. Then

\pi_2(n) = \sum_^pi(n/p_k) - k + 1 /math>
where \pi(x) is the

prime-counting function In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number ''x''. It is denoted by (''x'') (unrelated to the number ). History Of great interest in number theory is t ...

and

p_k

denotes the ''k''th prime.

Properties

Semiprime numbers have no

composite number A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime, ...

s as factors other than themselves. For example, the number 26 is semiprime and its only factors are 1, 2, 13, and 26, of which only 26 is composite. For a squarefree semiprime

n=pq

(with

p\ne q

) the value of

Euler's totient function In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ot ...

\varphi(n)

(the number of positive integers less than or equal to

n

that are

relatively prime In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...

n

) takes the simple form

\varphi(n)=(p-1)(q-1)=n-(p+q)+1.

This calculation is an important part of the application of semiprimes in the

RSA cryptosystem RSA (Rivest–Shamir–Adleman) is a public-key cryptosystem that is widely used for secure data transmission. It is also one of the oldest. The acronym "RSA" comes from the surnames of Ron Rivest, Adi Shamir and Leonard Adleman, who publicly ...

. For a square semiprime

n=p^2

, the formula is again simple:

\varphi(n)=p(p-1)=n-p.

Applications

Semiprimes are highly useful in the area of

cryptography Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adver ...

and

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777� ...

, most notably in

public key cryptography Public-key cryptography, or asymmetric cryptography, is the field of cryptographic systems that use pairs of related keys. Each key pair consists of a public key and a corresponding private key. Key pairs are generated with cryptographic alg ...

, where they are used by RSA and

pseudorandom number generator A pseudorandom number generator (PRNG), also known as a deterministic random bit generator (DRBG), is an algorithm for generating a sequence of numbers whose properties approximate the properties of sequences of random numbers. The PRNG-generate ...

s such as

Blum Blum Shub Blum Blum Shub (B.B.S.) is a pseudorandom number generator proposed in 1986 by Lenore Blum, Manuel Blum and Michael Shub that is derived from Michael O. Rabin's one-way function. __TOC__ Blum Blum Shub takes the form :x_ = x_n^2 \bmod M, where ...

. These methods rely on the fact that finding two large primes and multiplying them together (resulting in a semiprime) is computationally simple, whereas finding the original factors appears to be difficult. In the

RSA Factoring Challenge The RSA Factoring Challenge was a challenge put forward by RSA Laboratories on March 18, 1991 to encourage research into computational number theory and the practical difficulty of factoring large integers and cracking RSA keys used in cryptograp ...

RSA Security RSA Security LLC, formerly RSA Security, Inc. and doing business as RSA, is an American computer and network security company with a focus on encryption and encryption standards. RSA was named after the initials of its co-founders, Ron Rivest, ...

offered prizes for the factoring of specific large semiprimes and several prizes were awarded. The original RSA Factoring Challenge was issued in 1991, and was replaced in 2001 by the New RSA Factoring Challenge, which was later withdrawn in 2007. In 1974 the

Arecibo message The Arecibo message is an interstellar radio message carrying basic information about humanity and Earth that was sent to the globular cluster Messier 13 in 1974. It was meant as a demonstration of human technological achievement, rather than a ...

was sent with a radio signal aimed at a

star cluster Star clusters are large groups of stars. Two main types of star clusters can be distinguished: globular clusters are tight groups of ten thousand to millions of old stars which are gravitationally bound, while open clusters are more loosely clust ...

. It consisted of

1679

binary digits intended to be interpreted as a

23 \times 73

bitmap In computing, a bitmap is a mapping from some domain (for example, a range of integers) to bits. It is also called a bit array A bit array (also known as bitmask, bit map, bit set, bit string, or bit vector) is an array data structure that c ...

image. The number

1679=23\cdot 73

was chosen because it is a semiprime and therefore can be arranged into a rectangular image in only two distinct ways (23 rows and 73 columns, or 73 rows and 23 columns).

References

External links