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Newman–Shanks–Williams Prime
In mathematics, a Newman–Shanks–Williams prime (NSW prime) is a prime number ''p'' which can be written in the form :S_=\frac. NSW primes were first described by Morris Newman, Daniel Shanks and Hugh C. Williams in 1981 during the study of finite simple groups with square order. The first few NSW primes are 7, 41, 239, 9369319, 63018038201, … , corresponding to the indices 3, 5, 7, 19, 29, … . The sequence ''S'' alluded to in the formula can be described by the following recurrence relation: :S_0=1 \, :S_1=1 \, :S_n=2S_+S_\qquad\textn\geq 2. The first few terms of the sequence are 1, 1, 3, 7, 17, 41, 99, … . Each term in this sequence is half the corresponding term in the sequence of companion Pell numbers. These numbers also appear in the continued fraction In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow method of checking the primality of a given number n, called trial division, tests whether n is a multiple of any integer between 2 and \sqrt. Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, which always pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morris Newman
Morris may refer to: Places Australia * St Morris, South Australia, place in South Australia Canada * Morris Township, Ontario, now part of the municipality of Morris-Turnberry * Rural Municipality of Morris, Manitoba ** Morris, Manitoba, a town mostly surrounded by the municipality * Morris (electoral district), Manitoba (defunct) * Rural Municipality of Morris No. 312, Saskatchewan United States ;Communities * Morris, Alabama, a town * Morris, Connecticut, a town * Morris, Georgia, an unincorporated community * Morris, Illinois, a city * Morris, Indiana, an unincorporated community * Morris, Minnesota, a city * Morristown, New Jersey, a town * Morris (town), New York ** Morris (village), New York * Morris, Oklahoma, a city * Morris, Pennsylvania, an unincorporated community * Morris, West Virginia, an unincorporated community * Morris, Kanawha County, West Virginia, a ghost town * Morris, Wisconsin, a town * Morris Township (other) ;Counties and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Daniel Shanks
Daniel Shanks (January 17, 1917 – September 6, 1996) was an American mathematician who worked primarily in numerical analysis and number theory. He was the first person to compute π to 100,000 decimal places. Life and education Shanks was born on January 17, 1917, in Chicago, Illinois. He is not related to the English mathematician William Shanks, who was also known for his computation of π. He earned his Bachelor of Science degree in physics from the University of Chicago in 1937, and a Ph.D. in Mathematics from the University of Maryland in 1954. Prior to obtaining his PhD, Shanks worked at the Aberdeen Proving Ground and the Naval Ordnance Laboratory, first as a physicist and then as a mathematician. During this period he wrote his PhD thesis, which completed in 1949, despite having never taken any graduate math courses. After earning his PhD in mathematics, Shanks continued working at the Naval Ordnance Laboratory and the Naval Ship Research and Development Center at Da ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hugh C
Hugh may refer to: *Hugh (given name) Noblemen and clergy French * Hugh the Great (died 956), Duke of the Franks * Hugh Magnus of France (1007–1025), co-King of France under his father, Robert II * Hugh, Duke of Alsace (died 895), modern-day France * Hugh of Austrasia (7th century), Mayor of the Palace of Austrasia * Hugh I, Count of Angoulême (1183–1249) * Hugh II, Count of Angoulême (1221–1250) * Hugh III, Count of Angoulême (13th century) * Hugh IV, Count of Angoulême (1259–1303) * Hugh, Bishop of Avranches (11th century), France * Hugh I, Count of Blois (died 1248) * Hugh II, Count of Blois (died 1307) * Hugh of Brienne (1240–1296), Count of the medieval French County of Brienne * Hugh, Duke of Burgundy (d. 952) * Hugh I, Duke of Burgundy (1057–1093) * Hugh II, Duke of Burgundy (1084–1143) * Hugh III, Duke of Burgundy (1142–1192) * Hugh IV, Duke of Burgundy (1213–1272) * Hugh V, Duke of Burgundy (1294–1315) * Hugh Capet (939–996), King of France * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Simple Group
Finite is the opposite of infinite. It may refer to: * Finite number (other) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked for person and/or tense or aspect * "Finite", a song by Sara Groves from the album ''Invisible Empires'' See also * * Nonfinite (other) Nonfinite is the opposite of finite * a nonfinite verb A nonfinite verb is a derivative form of a verb unlike finite verbs. Accordingly, nonfinite verb forms are inflected for neither number nor person, and they cannot perform action as the root ... {{disambiguation fr:Fini it:Finito ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
7 (number)
7 (seven) is the natural number following 6 and preceding 8. It is the only prime number preceding a cube. As an early prime number in the series of positive integers, the number seven has greatly symbolic associations in religion, mythology, superstition and philosophy. The seven Classical planets resulted in seven being the number of days in a week. It is often considered lucky in Western culture and is often seen as highly symbolic. Unlike Western culture, in Vietnamese culture, the number seven is sometimes considered unlucky. It is the first natural number whose pronunciation contains more than one syllable. Evolution of the Arabic digit In the beginning, Indians wrote 7 more or less in one stroke as a curve that looks like an uppercase vertically inverted. The western Ghubar Arabs' main contribution was to make the longer line diagonal rather than straight, though they showed some tendencies to making the digit more rectilinear. The eastern Arabs developed the digit fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
41 (number)
41 (forty-one, XLI) is the natural number following 40 and preceding 42. In mathematics * the 13th smallest prime number. The next is 43, making both twin primes. * the sum of the first six prime numbers (2 + 3 + 5 + 7 + 11 + 13). * the 12th supersingular prime * a Newman–Shanks–Williams prime. * the smallest Sophie Germain prime to start a Cunningham chain of the first kind of three terms, . * an Eisenstein prime, with no imaginary part and real part of the form 3''n'' − 1. * a Proth prime as it is 5 × 23 + 1. * the largest lucky number of Euler: the polynomial yields primes for all the integers ''k'' with . * the sum of two squares, 42 + 52. * the sum of the sum of the divisors of the first 7 positive integers. * the smallest integer whose reciprocal has a 5-digit repetend. That is a consequence of the fact that 41 is a factor of 99999. * the smallest integer whose s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
239 (number)
239 (two hundred ndthirty-nine) is the natural number following 238 and preceding 240. In mathematics It is a prime number. The next is 241, with which it forms a pair of twin primes; hence, it is also a Chen prime. 239 is a Sophie Germain prime and a Newman–Shanks–Williams prime. It is an Eisenstein prime with no imaginary part and real part of the form 3''n'' − 1 (with no exponentiation implied). 239 is also a happy number. 239 is the smallest positive integer ''d'' such that the imaginary quadratic field Q() has class number = 15. HAKMEM (incidentally AI memo 239 of the MIT AI Lab) included an item on the properties of 239, including these: * When expressing 239 as a sum of square numbers, 4 squares are required, which is the maximum that any integer can require; it also needs the maximum number (9) of positive cubes (23 is the only other such integer), and the maximum number (19) of fourth powers. * 239/ 169 is a convergent of the continued fraction of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an ''arbitrary'' index set. For example, (M, A, R, Y) is a sequence of letters with the letter 'M' first and 'Y' last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be ''finite'', as in these examples, or ''infi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recurrence Relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter k that is independent of n; this number k is called the ''order'' of the relation. If the values of the first k numbers in the sequence have been given, the rest of the sequence can be calculated by repeatedly applying the equation. In ''linear recurrences'', the th term is equated to a linear function of the k previous terms. A famous example is the recurrence for the Fibonacci numbers, F_n=F_+F_ where the order k is two and the linear function merely adds the two previous terms. This example is a linear recurrence with constant coefficients, because the coefficients of the linear function (1 and 1) are constants that do not depend on n. For these recurrences, one can express the general term of the sequence as a closed-form expression o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pell Number
In mathematics, the Pell numbers are an infinite sequence of integers, known since ancient times, that comprise the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins , , , , and , so the sequence of Pell numbers begins with 1, 2, 5, 12, and 29. The numerators of the same sequence of approximations are half the companion Pell numbers or Pell–Lucas numbers; these numbers form a second infinite sequence that begins with 2, 6, 14, 34, and 82. Both the Pell numbers and the companion Pell numbers may be calculated by means of a recurrence relation similar to that for the Fibonacci numbers, and both sequences of numbers grow exponentially, proportionally to powers of the silver ratio 1 + . As well as being used to approximate the square root of two, Pell numbers can be used to find square triangular numbers, to construct integer approximations to the right isosceles triangle, and to solve certain combinat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
