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43,112,609 (number)
43,112,609 (forty-three million, one hundred twelve thousand, six hundred nine) is the natural number following 43,112,608 and preceding 43,112,610. In mathematics 43,112,609 is a prime number. Moreover, it is the exponent of the 47th Mersenne prime, equal to M43,112,609 = 243,112,609 − 1, a prime number with 12,978,189 decimal digits. It was discovered on August 23, 2008 by Edson Smith, a volunteer of the Great Internet Mersenne Prime Search. The 45th Mersenne prime, M37,156,667 = 237,156,667 − 1, was discovered two weeks later on September 6, 2008, marking the shortest chronological gap between discoveries of Mersenne primes since the formation of the online collaborative project in 1996. It was the first time since 1963 when two Mersenne primes were discovered less than 30 days apart from each other. Less than a year later, on June 4, 2009, the 46th Mersenne prime, M42,643,801 = 242,643,801 − 1, was discovered by Odd Magnar Strindmo, a GIMPS p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 numbers'', and numbers used for ordering are called ''ordinal numbers''. Natural numbers are sometimes used as labels, known as '' nominal numbers'', having none of the properties of numbers in a mathematical sense (e.g. sports jersey numbers). Some definitions, including the standard ISO 80000-2, begin the natural numbers with , corresponding to the non-negative integers , whereas others start with , corresponding to the positive integers Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, while in other writings, that term is used instead for the integers (including negative integers). The natural numbers form a set. Many other number sets are built by succ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mersenne Prime
In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. If is a composite number then so is . Therefore, an equivalent definition of the Mersenne primes is that they are the prime numbers of the form for some prime . The exponents which give Mersenne primes are 2, 3, 5, 7, 13, 17, 19, 31, ... and the resulting Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ... . Numbers of the form without the primality requirement may be called Mersenne numbers. Sometimes, however, Mersenne numbers are defined to have the additional requirement that be prime. The smallest composite Mersenne number with prime exponent ''n'' is . Mersenne primes were studied in antiquity because of their close connection to perfect numbers: the Euclid–Euler theorem as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Internet Mersenne Prime Search
The Great Internet Mersenne Prime Search (GIMPS) is a collaborative project of volunteers who use freely available software to search for Mersenne prime numbers. GIMPS was founded in 1996 by George Woltman, who also wrote the Prime95 client and its Linux port MPrime. Scott Kurowski wrote the back end PrimeNet server to demonstrate volunteer computing software by Entropia, a company he founded in 1997. GIMPS is registered as Mersenne Research, Inc. with Kurowski as Executive Vice President and board director. GIMPS is said to be one of the first large scale volunteer computing projects over the Internet for research purposes. , the project has found a total of seventeen Mersenne primes, fifteen of which were the largest known prime number at their respective times of discovery. The largest known prime is 282,589,933 − 1 (or M82,589,933 for short) and was discovered on December 7, 2018, by Patrick Laroche. On December 4, 2020, the project passed a major milestone aft ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Norway
Norway, officially the Kingdom of Norway, is a Nordic country in Northern Europe, the mainland territory of which comprises the western and northernmost portion of the Scandinavian Peninsula. The remote Arctic island of Jan Mayen and the archipelago of Svalbard also form part of Norway. Bouvet Island, located in the Subantarctic, is a dependency of Norway; it also lays claims to the Antarctic territories of Peter I Island and Queen Maud Land. The capital and largest city in Norway is Oslo. Norway has a total area of and had a population of 5,425,270 in January 2022. The country shares a long eastern border with Sweden at a length of . It is bordered by Finland and Russia to the northeast and the Skagerrak strait to the south, on the other side of which are Denmark and the United Kingdom. Norway has an extensive coastline, facing the North Atlantic Ocean and the Barents Sea. The maritime influence dominates Norway's climate, with mild lowland temperatures on the sea co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primitive Element (finite Field)
In field theory, a primitive element of a finite field is a generator of the multiplicative group of the field. In other words, is called a primitive element if it is a primitive th root of unity in ; this means that each non-zero element of can be written as for some integer . If is a prime number, the elements of can be identified with the integers modulo . In this case, a primitive element is also called a primitive root modulo . For example, 2 is a primitive element of the field and , but not of since it generates the cyclic subgroup of order 3; however, 3 is a primitive element of . The minimal polynomial of a primitive element is a primitive polynomial. Properties Number of primitive elements The number of primitive elements in a finite field is , where is Euler's totient function, which counts the number of elements less than or equal to which are relatively prime to . This can be proved by using the theorem that the multiplicative group of a finite fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trinomial
In elementary algebra, a trinomial is a polynomial consisting of three terms or monomials. Examples of trinomial expressions # 3x + 5y + 8z with x, y, z variables # 3t + 9s^2 + 3y^3 with t, s, y variables # 3ts + 9t + 5s with t, s variables # ax^2+bx+c, the quadratic expression in standard form with a,b,c variables. # A x^a y^b z^c + B t + C s with x, y, z, t, s variables, a, b, c nonnegative integers and A, B, C any constants. # Px^a + Qx^b + Rx^c where x is variable and constants a, b, c are nonnegative integers and P, Q, R any constants. Trinomial equation A trinomial equation is a polynomial equation involving three terms. An example is the equation x = q + x^m studied by Johann Heinrich Lambert in the 18th century. Some notable trinomials * The quadratic trinomial in standard form (as from above): ax^2+bx+c See also *Trinomial expansion *Monomial *Binomial * Multinomial *Simple expression In mathematics, a monomial is, roughly speaking, a polynomial which has on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
GF(2)
(also denoted \mathbb F_2, or \mathbb Z/2\mathbb Z) is the finite field of two elements (GF is the initialism of ''Galois field'', another name for finite fields). Notations and \mathbb Z_2 may be encountered although they can be confused with the notation of -adic integers. is the field with the smallest possible number of elements, and is unique if the additive identity and the multiplicative identity are denoted respectively and , as usual. The elements of may be identified with the two possible values of a bit and to the boolean values ''true'' and ''false''. It follows that is fundamental and ubiquitous in computer science and its logical foundations. Definition GF(2) is the unique field with two elements with its additive and multiplicative identities respectively denoted and . Its addition is defined as the usual addition of integers but modulo 2 and corresponds to the table below: If the elements of GF(2) are seen as boolean values, then the addition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sophie Germain Prime
In number theory, a prime number ''p'' is a if 2''p'' + 1 is also prime. The number 2''p'' + 1 associated with a Sophie Germain prime is called a . For example, 11 is a Sophie Germain prime and 2 × 11 + 1 = 23 is its associated safe prime. Sophie Germain primes are named after French mathematician Sophie Germain, who used them in her investigations of Fermat's Last Theorem. One attempt by Germain to prove Fermat’s Last Theorem was to let ''p'' be a prime number of the form 8''k'' + 7 and to let ''n'' = ''p'' – 1. In this case, x^n + y^n = z^n is unsolvable. Germain’s proof, however, remained unfinished. Through her attempts to solve Fermat's Last Theorem, Germain developed a result now known as Germain's Theorem which states that if ''p'' is an odd prime and 2''p'' + 1 is also prime, then ''p'' must divide ''x'', ''y'', or ''z.'' Otherwise, x^n + y^n \neq z^n. This case where ''p'' does not divide ''x'', ''y'', or ''z'' i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaussian Prime
In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as \mathbf /math> or \Z Gaussian integers share many properties with integers: they form a Euclidean domain, and have thus a Euclidean division and a Euclidean algorithm; this implies unique factorization and many related properties. However, Gaussian integers do not have a total ordering that respects arithmetic. Gaussian integers are algebraic integers and form the simplest ring of quadratic integers. Gaussian integers are named after the German mathematician Carl Friedrich Gauss. Basic definitions The Gaussian integers are the set :\mathbf \, \qquad \text i^2 = -1. In other words, a Gaussian integer is a complex number such that its real and imaginary parts are both integers. Since the Gaussian integers are closed under addition and multip ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integers
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of mathematics, the set of integers is often denoted by the boldface or blackboard bold \mathbb. The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the natural numbers, \mathbb is countably infinite. An integer may be regarded as a real number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and are not. The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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