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Biprime
In mathematics, a semiprime is a natural number that is the product of exactly two prime numbers. The two primes in the product may equal each other, so the semiprimes include the squares of prime numbers. Because there are infinitely many prime numbers, there are also infinitely many semiprimes. Semiprimes are also called biprimes, since they include two primes, or second numbers, by analogy with how "prime" means "first". Alternatively non-prime semiprimes are called almost-prime numbers, specifically the "2-almost-prime" biprime and "3-almost-prime" triprime 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 primes, 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: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
RSA (algorithm)
The RSA (Rivest–Shamir–Adleman) cryptosystem is a public-key cryptosystem, one of the oldest widely used for secure data transmission. The initialism "RSA" comes from the surnames of Ron Rivest, Adi Shamir and Leonard Adleman, who publicly described the algorithm in 1977. An equivalent system was developed secretly in 1973 at Government Communications Headquarters (GCHQ), the British signals intelligence agency, by the English mathematician Clifford Cocks. That system was declassified in 1997. In a public-key cryptosystem, the encryption key is public and distinct from the decryption key, which is kept secret (private). An RSA user creates and publishes a public key based on two large prime numbers, along with an auxiliary value. The prime numbers are kept secret. Messages can be encrypted by anyone via the public key, but can only be decrypted by someone who knows the private key. The security of RSA relies on the practical difficulty of factoring the product of two ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer Sequences
In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers. An integer sequence may be specified ''explicitly'' by giving a formula for its ''n''th term, or ''implicitly'' by giving a relationship between its terms. For example, the sequence 0, 1, 1, 2, 3, 5, 8, 13, ... (the Fibonacci sequence) is formed by starting with 0 and 1 and then adding any two consecutive terms to obtain the next one: an implicit description . The sequence 0, 3, 8, 15, ... is formed according to the formula ''n''2 − 1 for the ''n''th term: an explicit definition. Alternatively, an integer sequence may be defined by a property which members of the sequence possess and other integers do not possess. For example, we can determine whether a given integer is a perfect number, , even though we do not have a formula for the ''n''th perfect number. Computable and definable sequences An integer sequence is computable if there exists an algorithm that, given '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parity Problem (sieve Theory)
In number theory, the parity problem refers to a limitation in sieve theory that prevents sieves from giving good estimates in many kinds of prime-counting problems. The problem was identified and named by Atle Selberg in 1949. Beginning around 1996, John Friedlander and Henryk Iwaniec developed some parity-sensitive sieves that make the parity problem less of an obstacle. Statement Terence Tao gave this "rough" statement of the problem: This problem is significant because it may explain why it is difficult for sieves to "detect primes," in other words to give a non-trivial lower bound for the number of primes with some property. For example, in a sense Chen's theorem is very close to a solution of the twin prime conjecture, since it says that there are infinitely many primes ''p'' such that ''p'' + 2 is either prime or the product of two primes (semiprime). The parity problem suggests that, because the case of interest has an odd number of prime factors (namely 1), it won't be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sphenic Number
In number theory, a sphenic number (from , 'wedge') is a positive integer that is the product of three distinct prime numbers. Because there are infinitely many prime numbers, there are also infinitely many sphenic numbers. Definition A sphenic number is a product ''pqr'' where ''p'', ''q'', and ''r'' are three distinct prime numbers. In other words, the sphenic numbers are the square-free 3- almost primes. Examples The smallest sphenic number is 30 = 2 × 3 × 5, the product of the smallest three primes. The first few sphenic numbers are : 30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, ... The largest known sphenic number at any time can be obtained by multiplying together the three largest known primes. Divisors All sphenic numbers have exactly eight divisors. If we express the sphenic number as n = p \cdot q \cdot r, where ''p'', ''q'', and ''r'' are distinct primes, then the set of divisors of ''n'' will be: :\left\. The converse does not hold. F ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chen's Theorem
In number theory, Chen's theorem states that every sufficiently large parity (mathematics), even number can be written as the sum of either two prime number, primes, or a prime and a semiprime (the product of two primes). It is a weakened form of Goldbach's conjecture, which states that every even number is the sum of two primes. History The theorem was first stated by China, Chinese mathematician Chen Jingrun in 1966, with further details of the mathematical proof, proof in 1973. His original proof was much simplified by P. M. Ross in 1975. Chen's theorem is a significant step towards Goldbach's conjecture, and a celebrated application of sieve theory, sieve methods. Chen's theorem represents the strengthening of a previous result due to Alfréd Rényi, who in 1947 had shown there exists a finite ''K'' such that any even number can be written as the sum of a prime number and the product of at most ''K'' primes. Variations Chen's 1973 paper stated two results with nearly i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bitmap
In computing, a bitmap (also called raster) graphic is an image formed from rows of different colored pixels. A GIF is an example of a graphics image file that uses a bitmap. As a noun, the term "bitmap" is very often used to refer to a particular bitmapping application: the pix-map, which refers to a map of pixels, where each pixel may store more than two colors, thus using more than one bit per pixel. In such a case, the domain in question is the array of pixels which constitute a digital graphic output device (a screen or monitor). In some contexts, the term ''bitmap'' implies one bit per pixel, whereas ''pixmap'' is used for images with multiple bits per pixel. A bitmap is a type of memory organization or image file format used to store digital images. The term ''bitmap'' comes from the computer programming terminology, meaning just a ''map of bits'', a spatially mapped array of bits. Now, along with ''pixmap'', it commonly refers to the similar concept of a spatially mapp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Star Cluster
A star cluster is a group of stars held together by self-gravitation. Two main types of star clusters can be distinguished: globular clusters, tight groups of ten thousand to millions of old stars which are gravitationally bound; and open clusters, less tight groups of stars, generally containing fewer than a few hundred members. As they move through the galaxy, over time, open clusters become disrupted by the gravitational influence of giant molecular clouds, so that the clusters we observe are often young. Even though they are no longer gravitationally bound, they will continue to move in broadly the same direction through space and are then known as stellar associations, sometimes referred to as ''moving groups''. Globular clusters, with more members and more mass, remain intact for far longer and the globular clusters we observe are usually billions of years old. Star clusters visible to the naked eye include the Pleiades and Hyades (star cluster), Hyades open clusters, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 real attempt to enter into a conversation with extraterrestrials. The message was broadcast into space a single time via frequency modulated radio waves at a ceremony to mark the remodeling of the Arecibo Telescope in Puerto Rico on 16 November 1974. The message was aimed at the current location of M13, about 25,000 light years from Earth, because M13 was a large and relatively close collection of stars that was available in the sky at the time and place of the ceremony. When correctly translated into graphics, characters, and spaces, the 1,679 bits of data contained within the message form the image shown here. Description The content of the Arecibo message was designed by a group of Cornell University and Arecibo scientists: Frank ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
RSA Security
RSA Security LLC, formerly RSA Security, Inc. and trade name RSA, is an American computer security, computer and network security company with a focus on encryption and decryption standards. RSA was named after the initials of its co-founders, Ron Rivest, Adi Shamir and Leonard Adleman, after whom the RSA (algorithm), RSA public key cryptography algorithm was also named. Among its products is the SecurID authentication token. The BSAFE cryptography libraries were also initially owned by RSA. RSA is known for incorporating backdoors developed by the National Security Agency, NSA in its products. It also organizes the annual RSA Conference, an information security conference. Founded as an independent company in 1982, RSA Security was acquired by EMC Corporation in 2006 for US$2.1 billion and operated as a division within EMC. When EMC was acquired by Dell Technologies in 2016, RSA became part of the Dell Technologies family of brands. On 10 March 2020, Dell Technologies announced ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 cryptography. They published a list of semiprimes (numbers with exactly two prime factors) known as the RSA numbers, with a cash prize for the successful factorization of some of them. The smallest of them, a 100-decimal digit number called RSA-100 was factored by April 1, 1991. Many of the bigger numbers have still not been factored and are expected to remain unfactored for quite some time, however advances in quantum computers make this prediction uncertain due to Shor's algorithm. In 2001, RSA Laboratories expanded the factoring challenge and offered prizes ranging from $10,000 to $200,000 for factoring numbers from 576 bits up to 2048 bits. The RSA Factoring Challenges ended in 2007. RSA Laboratories stated: "Now that the industry has a cons ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer Factorization
In mathematics, integer factorization is the decomposition of a positive integer into a product of integers. Every positive integer greater than 1 is either the product of two or more integer factors greater than 1, in which case it is a composite number, or it is not, in which case it is a prime number. For example, is a composite number because , but is a prime number because it cannot be decomposed in this way. If one of the factors is composite, it can in turn be written as a product of smaller factors, for example . Continuing this process until every factor is prime is called prime factorization; the result is always unique up to the order of the factors by the prime factorization theorem. To factorize a small integer using mental or pen-and-paper arithmetic, the simplest method is trial division: checking if the number is divisible by prime numbers , , , and so on, up to the square root of . For larger numbers, especially when using a computer, various more sophis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
