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Ulam Spiral
The Ulam spiral or prime spiral is a graphical depiction of the set of prime numbers, devised by mathematician Stanisław Ulam in 1963 and popularized in Martin Gardner's ''Mathematical Games'' column in ''Scientific American'' a short time later. It is constructed by writing the positive integers in a square spiral and specially marking the prime numbers. Ulam and Gardner emphasized the striking appearance in the spiral of prominent diagonal, horizontal, and vertical lines containing large numbers of primes. Both Ulam and Gardner noted that the existence of such prominent lines is not unexpected, as lines in the spiral correspond to quadratic polynomials, and certain such polynomials, such as Euler's prime-generating polynomial ''x''2 − ''x'' + 41, are believed to produce a high density of prime numbers. Nevertheless, the Ulam spiral is connected with major unsolved problems in number theory such as Landau's problems. In particular, no quadratic polynomial has ever ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ulam Spiral 10x
Ulam may refer to: * ULAM, the ICAO airport code for Naryan-Mar Airport, Russia * Ulam (surname) * Ulam (salad), a type of Malay salad * ''Ulam'', a Filipino term loosely translated to viand or side dish; see Tapa (Filipino cuisine) * Ulam, the language spoken by prehistoric humans in the movie ''Quest for Fire (film), Quest for Fire'' * Ulam, the name of the porch of Solomon's Temple held up by the pillars *Thalassognathus ulami, a Marine vertebrate from the phylum, Chordata, and class, Actinopterygii (ray-finned fishes). The Tribe - Local Name is Ulam (from Tagalog, meaning food or meat dish). It resembles a mix between an anglerfish and a barracuda. See also *List of things named after Stanislaw Ulam * Anglerfish * Barracuda * Deep-sea fish {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Lattice
In mathematics, the square lattice is a type of lattice in a two-dimensional Euclidean space. It is the two-dimensional version of the integer lattice, denoted as . It is one of the five types of two-dimensional lattices as classified by their symmetry groups; its symmetry group in IUC notation as , Coxeter notation as , and orbifold notation as . Two orientations of an image of the lattice are by far the most common. They can conveniently be referred to as the upright square lattice and diagonal square lattice; the latter is also called the centered square lattice.. They differ by an angle of 45°. This is related to the fact that a square lattice can be partitioned into two square sub-lattices, as is evident in the colouring of a checkerboard. Symmetry The square lattice's symmetry category is wallpaper group . A pattern with this lattice of translational symmetry cannot have more, but may have less symmetry than the lattice itself. An upright square lattice can be vi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number Theorem
In mathematics, the prime number theorem (PNT) describes the asymptotic analysis, asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs. The theorem was proved independently by Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896 using ideas introduced by Bernhard Riemann (in particular, the Riemann zeta function). The first such distribution found is , where is the prime-counting function (the number of primes less than or equal to ''N'') and is the natural logarithm of . This means that for large enough , the probability that a random integer not greater than is prime is very close to . Consequently, a random integer with at most digits (for large enough ) is about half as likely to be prime as a random integer with at most digits. For example, among the positive integers of at most 1000 digits, about on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bunyakovsky Conjecture
The Bunyakovsky conjecture (or Bouniakowsky conjecture) gives a criterion for a polynomial f(x) in one variable with integer coefficients to give infinitely many prime values in the sequencef(1), f(2), f(3),\ldots. It was stated in 1857 by the Russian mathematician Viktor Bunyakovsky. The following three conditions are necessary for f(x) to have the desired prime-producing property: # The leading coefficient is positive, # The polynomial is irreducible over the rationals (and integers), and # There is no common factor for all the infinitely many values f(1), f(2), f(3),\ldots. (In particular, the coefficients of f(x) should be relatively prime. It is not necessary for the values f(n) to be pairwise relatively prime.) Bunyakovsky's conjecture is that these conditions are sufficient: if f(x) satisfies (1)–(3), then f(n) is prime for infinitely many positive integers n. A seemingly weaker yet equivalent statement to Bunyakovsky's conjecture is that for every integer polyno ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Composite Numbers
A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Accordingly it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime, or the unit 1, so the composite numbers are exactly the numbers that are not prime and not a unit. E.g., the integer 14 is a composite number because it is the product of the two smaller integers 2 × 7 but the integers 2 and 3 are not because each can only be divided by one and itself. The composite numbers up to 150 are: :4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 128, 129 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Number
In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals and can be written as . The usual notation for the square of a number is not the product , but the equivalent exponentiation , usually pronounced as " squared". The name ''square'' number comes from the name of the shape. The unit of area is defined as the area of a unit square (). Hence, a square with side length has area . If a square number is represented by ''n'' points, the points can be arranged in rows as a square each side of which has the same number of points as the square root of ''n''; thus, square numbers are a type of Figurate number, figurate numbers (other examples being Cube (algebra), cube numbers and triangular numbers). In the Real number, real number system, square numbers are non-negative. A non-negative integer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ulam 2
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Ulam may refer to: * ULAM, the ICAO airport code for Naryan-Mar Airport, Russia * Ulam (surname) * Ulam (salad), a type of Malay salad * ''Ulam'', a Filipino term loosely translated to viand or side dish; see Tapa (Filipino cuisine) * Ulam, the language spoken by prehistoric humans in the movie '' Quest for Fire'' * Ulam, the name of the porch of Solomon's Temple held up by the pillars *Thalassognathus ulami, a Marine vertebrate from the phylum, Chordata, and class, Actinopterygii (ray-finned fishes). The Tribe - Local Name is Ulam (from Tagalog, meaning food or meat dish). It resembles a mix between an anglerfish and a barracuda. See also * List of things named after Stanislaw Ulam * Anglerfish * Barracuda * Deep-sea fish Deep-sea fish are fish that live in the darkness below the sunlit surface waters, that is below the epipelagic or photic zone of the sea. The lanternfish is, by far, the most common deep-sea fish. Other deep-sea fishes include the flashlight f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discriminant
In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the zero of a function, roots without computing them. More precisely, it is a polynomial function of the coefficients of the original polynomial. The discriminant is widely used in polynomial factorization, polynomial factoring, number theory, and algebraic geometry. The discriminant of the quadratic polynomial ax^2+bx+c is :b^2-4ac, the quantity which appears under the square root in the quadratic formula. If a\ne 0, this discriminant is zero if and only if the polynomial has a double root. In the case of real number, real coefficients, it is positive if the polynomial has two distinct real roots, and negative if it has two distinct complex conjugate roots. Similarly, the discriminant of a cubic polynomial is zero if and only if the polynomial has a multiple root. In the case of a cubic with real coefficients, the discriminant is positive if the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bateman–Horn Conjecture
In number theory, the Bateman–Horn conjecture is a statement concerning the frequency of prime numbers among the values of a system of polynomials, named after mathematicians Paul T. Bateman and Roger A. Horn who proposed it in 1962. It provides a vast generalization of such conjectures as the Hardy and Littlewood conjecture on the density of twin primes or their conjecture on primes of the form ''n''2 + 1; it is also a strengthening of Schinzel's hypothesis H. Definition The Bateman–Horn conjecture provides a conjectured density for the positive integers at which a given set of polynomials all have prime values. For a set of ''m'' distinct irreducible polynomials ''ƒ''1, ..., ''ƒ''''m'' with integer coefficients, an obvious necessary condition for the polynomials to simultaneously generate prime values infinitely often is that they satisfy Bunyakovsky's property, that there does not exist a prime number ''p'' that divides their product ''f' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Edensor Littlewood
John Edensor Littlewood (9 June 1885 – 6 September 1977) was a British mathematician. He worked on topics relating to analysis, number theory, and differential equations and had lengthy collaborations with G. H. Hardy, Srinivasa Ramanujan and Mary Cartwright. Biography Littlewood was born on the 9th of June 1885 in Rochester, Kent, the eldest son of Edward Thornton Littlewood and Sylvia Maud (née Ackland). In 1892, his father accepted the headmastership of a school in Wynberg, Cape Town, in South Africa, taking his family there. Littlewood returned to Britain in 1900 to attend St Paul's School in London, studying under Francis Sowerby Macaulay, an influential algebraic geometer. In 1903, Littlewood entered the University of Cambridge, studying in Trinity College. He spent his first two years preparing for the Tripos examinations which qualify undergraduates for a bachelor's degree where he emerged in 1905 as Senior Wrangler bracketed with James Mercer (Mercer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Goldbach's Conjecture
Goldbach's conjecture is one of the oldest and best-known list of unsolved problems in mathematics, unsolved problems in number theory and all of mathematics. It states that every even and odd numbers, even natural number greater than 2 is the sum of two prime numbers. The conjecture has been shown to hold for all integers less than but remains unproven despite considerable effort. History Origins On 7 June 1742, the Prussian mathematician Christian Goldbach wrote a letter to Leonhard Euler (letter XLIII), in which he proposed the following conjecture: Goldbach was following the now-abandoned convention of Prime number#Primality of one, considering 1 to be a prime number, so that a sum of units would be a sum of primes. He then proposed a second conjecture in the margin of his letter, which implies the first: Euler replied in a letter dated 30 June 1742 and reminded Goldbach of an earlier conversation they had had (""), in which Goldbach had remarked that the first of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Los Alamos National Laboratory
Los Alamos National Laboratory (often shortened as Los Alamos and LANL) is one of the sixteen research and development Laboratory, laboratories of the United States Department of Energy National Laboratories, United States Department of Energy (DOE), located a short distance northwest of Santa Fe, New Mexico, in the Southwestern United States, American southwest. Best known for its central role in helping develop the First Atomic bomb, first atomic bomb, LANL is one of the world's largest and most advanced scientific institutions. Los Alamos was established in 1943 as Project Y, a top-secret site for designing nuclear weapons under the Manhattan Project during World War II.The site was variously called Los Alamos Laboratory and Los Alamos Scientific Laboratory. Chosen for its remote yet relatively accessible location, it served as the main hub for conducting and coordinating nuclear research, bringing together some of the world's most famous scientists, among them numerous Nobel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
