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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Number theory, a Natural number is called k -almost prime If and only if it has exactly k Prime factors counted with In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 The first few semiprimes are 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, . . . (sequence A001358 in OEIS). The On-Line Encyclopedia of Integer Sequences ( OEIS) also cited simply as Sloane's, is an extensive searchable Database of Integer sequences
As of 2007, the largest known semiprime is (232,582,657 − 1)2, which has over 19 million digits. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. This is the square of the largest known prime. In Algebra, the square of a number is that number multiplied by itself 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 that somebody could find a way to prove a larger number is a semiprime without knowing the two factors, but so far that has only happened for smaller semiprimes. [1]
The value of Euler's totient function for a semiprime n = pq is particularly simple when p and q are distinct:
- φ(n) = n + 1 − (p + q). In Number theory, the totient \varphi(n of a Positive integer n is defined to be the number of positive integers less than or equal to
Contents |
Applications
Semiprimes are highly useful in the area of cryptography and number theory, most notably in public key cryptography, where they are used by RSA and pseudorandom number generators such as Blum Blum Shub. Cryptography (or cryptology; from Greek grc κρυπτός kryptos, "hidden secret" and grc γράφω gráphō, "I write" Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes Public-key cryptography, also known as asymmetric cryptography, is a form of Cryptography in which the key used to encrypt a message differs from the key In Cryptography, RSA is an Algorithm for Public-key cryptography. A pseudorandom number generator ( PRNG) is an Algorithm for generating a sequence of numbers that approximates the properties of random numbers Blum Blum Shub ( BBS) is a Pseudorandom number generator proposed in 1986 by Lenore Blum, Manuel Blum and Michael Shub (Blum et These methods rely on the fact that finding two large primes and multiplying them together is computationally simple, whereas finding the original factors appears to be difficult. In the RSA Factoring Challenge, RSA Security offered prizes for the factoring of specific large semiprimes and several prizes were awarded. The RSA Factoring Challenge was a challenge put forward by RSA Laboratories on March 18 1991 to encourage research into Computational number theory RSA The Security Division of EMC Corporation, is headquartered in Bedford Massachusetts, United States and maintains offices in Ireland, Israel The challenge closed in 2007. [1]
In practical cryptography, it is not sufficient to choose just any semiprime; a good number must evade a number of well-known special-purpose algorithms that can factor numbers of certain form. The factors p and q of n should be very large, around the same order of magnitude as the square root; this makes trial division and Pollard's rho algorithm impractical. Trial division is the most laborious but easiest to understand of the Integer factorization algorithms Pollard's rho algorithm is a special-purpose Integer factorization Algorithm. At the same time they cannot be too close together, or else another simple test can factor the number. The number may also be chosen so that none of p − 1, p + 1, q − 1, or q + 1 are smooth numbers, protecting against Pollard's p − 1 algorithm or Williams' p + 1 algorithm. In Number theory, a positive Integer is called B - smooth if none of its Prime factors are greater than B. Pollard's p &minus 1 algorithm is a number theoretic Integer factorization Algorithm, invented by John Pollard in 1974 In Computational number theory, Williams' p + 1 algorithm is an Integer factorization algorithm one of the family of Algebraic-group factorisation These checks cannot take future algorithms or secret algorithms into account however, introducing the possibility that numbers in use today may be broken by special-purpose algorithms.
In 1974 the Arecibo message was sent with a radio signal aimed at a star cluster. The Arecibo message was beamed via frequency modulated Radio waves into space at a ceremony to mark the remodeling of the Arecibo radio telescope Star clusters are groups of Stars which are gravitationally bound It consisted of 1679 binary digits intended to be interpreted as a 23×73 bitmap image. In Computer graphics, a bitmap or pixmap is a type of memory organization or Image file format used to store Digital images The The number 1679 = 23×73 was chosen because it is a semiprime and therefore can only be broken down into 23 rows and 73 columns, or 73 rows and 23 columns.
See also
References
- ^ Chris Caldwell, The Prime Glossary: semiprime at The Prime Pages. Chen's theorem was first stated by Chinese Mathematician Chen Jingrun in 1966 with further details of the proof in 1973 The Prime Pages is a website about Prime numbers maintained by Prof Retrieved on 2007-12-04. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. "December 4th" redirects here For the song by Jay-Z, see December 4th (song.
External links
- Eric W. Weisstein, Semi prime at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
Dapyx Software network: MP3 Explorer | Ebook Manager | Zenithic