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Star Of David Theorem
The Star of David theorem is a mathematical result on arithmetic properties of binomial coefficients. It was discovered by Henry W. Gould in 1972. Statement The greatest common divisors of the binomial coefficients forming each of the two triangles in the Star of David shape in Pascal's triangle are equal: : \begin & \gcd\left\ \\ pt= & \gcd\left\. \end Examples Rows 8, 9, and 10 of Pascal's triangle are For ''n''=9, ''k''=3 or ''n''=9, ''k''=6, the element 84 (circled bold) is surrounded by, in sequence, the elements 28, 56, 126, 210, 120 and 36 (bold). Taking alternating values, we have gcd(28, 126, 120) = 2 = gcd(56, 210, 36). The element 36 (circled italics) is surrounded by the sequence 8, 28, 84, 120, 45 and 9 (italics), and taking alternating values we have gcd(8, 84, 45) = 1 = gcd(28, 120, 9). Generalization The above greatest common divisor also equals \gcd \left(, , , \right). Thus in the above example for the element 84 (in its rightmost appearance), w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pascal Triangle Small
Pascal, Pascal's or PASCAL may refer to: People and fictional characters * Pascal (given name), including a list of people with the name * Pascal (surname), including a list of people and fictional characters with the name ** Blaise Pascal, French mathematician, physicist, inventor, philosopher, writer and theologian Places * Pascal (crater), a lunar crater * Pascal Island (Antarctica) * Pascal Island (Western Australia) Science and technology * Pascal (unit), the SI unit of pressure * Pascal (programming language), a programming language developed by Niklaus Wirth ** Microsoft Pascal **Turbo Pascal * PASCAL (database), a bibliographic database maintained by the Institute of Scientific and Technical Information * Pascal (microarchitecture), codename for a microarchitecture developed by Nvidia Other uses * (1895–1911) * (1931–1942) * Pascal and Maximus, fictional characters in ''Tangled'' * Pascal blanc, a French white wine grape * Pascal College, secondary education scho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic
Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms. Arithmetic systems can be distinguished based on the type of numbers they operate on. Integer arithmetic is about calculations with positive and negative integers. Rational number arithmetic involves operations on fractions of integers. Real number arithmetic is about calculations with real numbers, which include both rational and irrational numbers. Another distinction is based on the numeral system employed to perform calculations. Decimal arithmetic is the most common. It uses the basic numerals from 0 to 9 and their combinations to express numbers. Binary arithmetic, by contrast, is used by most computers and represents numbers as combinations of the basic numerals 0 and 1. Computer arithmetic deals with the specificities of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Coefficients
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the term in the polynomial expansion of the binomial power ; this coefficient can be computed by the multiplicative formula : \binom nk = \frac, which using factorial notation can be compactly expressed as : \binom = \frac. For example, the fourth power of is : \begin (1 + x)^4 &= \tbinom x^0 + \tbinom x^1 + \tbinom x^2 + \tbinom x^3 + \tbinom x^4 \\ &= 1 + 4x + 6 x^2 + 4x^3 + x^4, \end and the binomial coefficient \tbinom =\tfrac = \tfrac = 6 is the coefficient of the term. Arranging the numbers \tbinom, \tbinom, \ldots, \tbinom in successive rows for gives a triangular array called Pascal's triangle, satisfying the recurrence relation : \binom = \binom + \binom . The binomial coefficients occur in many areas of mathematics, and especia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Henry W
Henry may refer to: People and fictional characters * Henry (given name), including lists of people and fictional characters * Henry (surname) * Henry, a stage name of François-Louis Henry (1786–1855), French baritone Arts and entertainment * ''Henry'' (2011 film), a Canadian short film * ''Henry'' (2015 film), a virtual reality film * '' Henry: Portrait of a Serial Killer'', a 1986 American crime film * ''Henry'' (comics), an American comic strip created in 1932 by Carl Anderson * "Henry", a song by New Riders of the Purple Sage Places Antarctica * Henry Bay, Wilkes Land Australia * Henry River (New South Wales) * Henry River (Western Australia) Canada * Henry Lake (Vancouver Island), British Columbia * Henry Lake (Halifax County), Nova Scotia * Henry Lake (District of Chester), Nova Scotia New Zealand * Lake Henry (New Zealand) * Henry River (New Zealand) United States * Henry, Illinois * Henry, Indiana * Henry, Nebraska * Henry, South Dakota * Henry Count ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greatest Common Divisor
In mathematics, the greatest common divisor (GCD), also known as greatest common factor (GCF), of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers , , the greatest common divisor of and is denoted \gcd (x,y). For example, the GCD of 8 and 12 is 4, that is, . In the name "greatest common divisor", the adjective "greatest" may be replaced by "highest", and the word "divisor" may be replaced by "factor", so that other names include highest common factor, etc. Historically, other names for the same concept have included greatest common measure. This notion can be extended to polynomials (see ''Polynomial greatest common divisor'') and other commutative rings (see ' below). Overview Definition The ''greatest common divisor'' (GCD) of integers and , at least one of which is nonzero, is the greatest positive integer such that is a divisor of both and ; that is, there are integers and such that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Star Of David
The Star of David (, , ) is a symbol generally recognized as representing both Jewish identity and Judaism. Its shape is that of a hexagram: the compound of two equilateral triangles. A derivation of the Seal of Solomon was used for decorative and mystical purposes by Kabbalah, Kabbalistic Jews and Muslims. The hexagram appears occasionally in Jewish contexts since antiquity as a decorative motif, such as a stone bearing a hexagram from the arch of the 3rd–4th century Khirbet Shura synagogue. A hexagram found in a religious context can be seen in a Leningrad Codex, manuscript of the Hebrew Bible from 11th-century Cairo. Its association as a distinctive symbol for the Jewish people and their religion dates to 17th-century Prague. In the 19th century, the symbol began to be widely used by the History of the Jews in Europe, Jewish communities of Eastern Europe, ultimately coming to represent Jewish identity or religious beliefs."The Flag and the Emblem" (MFA). It became repr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pascal's Triangle
In mathematics, Pascal's triangle is an infinite triangular array of the binomial coefficients which play a crucial role in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in Persia, India, China, Germany, and Italy. The rows of Pascal's triangle are conventionally enumerated starting with row n = 0 at the top (the 0th row). The entries in each row are numbered from the left beginning with k = 0 and are usually staggered relative to the numbers in the adjacent rows. The triangle may be constructed in the following manner: In row 0 (the topmost row), there is a unique nonzero entry 1. Each entry of each subsequent row is constructed by adding the number above and to the left with the number above and to the right, treating blank entries as 0. For example, the initial number of row 1 (or any other row) is 1 (the sum of 0 and 1), whereas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Factorial And Binomial Topics
{{Short description, none This is a list of factorial and binomial topics in mathematics. See also binomial (other). * Abel's binomial theorem *Alternating factorial *Antichain *Beta function * Bhargava factorial *Binomial coefficient **Pascal's triangle *Binomial distribution *Binomial proportion confidence interval * Binomial-QMF ( Daubechies wavelet filters) *Binomial series *Binomial theorem *Binomial transform *Binomial type * Carlson's theorem *Catalan number ** Fuss–Catalan number *Central binomial coefficient *Combination *Combinatorial number system *De Polignac's formula *Difference operator * Difference polynomials *Digamma function * Egorychev method * Erdős–Ko–Rado theorem *Euler–Mascheroni constant *Faà di Bruno's formula *Factorial * Factorial moment *Factorial number system *Factorial prime * Factoriangular number *Gamma distribution *Gamma function *Gaussian binomial coefficient * Gould's sequence * Hyperfactorial *Hypergeometric distribution * Hy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fibonacci Quarterly
The ''Fibonacci Quarterly'' is a scientific journal on mathematical topics related to the Fibonacci numbers, published four times per year. It is the primary publication of The Fibonacci Association, which has published it since 1963. Its founding editors were Verner Emil Hoggatt Jr. and Alfred Brousseau; by Clark Kimberling the present editor is Professor Curtis Cooper of the Mathematics Department of the University of Central Missouri. The ''Fibonacci Quarterly'' has an editorial board of nineteen members and is overseen by the nine-me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematica
Wolfram (previously known as Mathematica and Wolfram Mathematica) is a software system with built-in libraries for several areas of technical computing that allows machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimization, plotting functions and various types of data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other programming languages. It was conceived by Stephen Wolfram, and is developed by Wolfram Research of Champaign, Illinois. The Wolfram Language is the programming language used in ''Mathematica''. Mathematica 1.0 was released on June 23, 1988 in Champaign, Illinois and Santa Clara, California. Mathematica's Wolfram Language is fundamentally based on Lisp; for example, the Mathematica command Most is identically equal to the Lisp command butlast. There is a substantial literature on the development of computer algebra systems (CAS). __TOC_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1972 Introductions
Year 197 ( CXCVII) was a common year starting on Saturday of the Julian calendar. At the time, it was known as the Year of the Consulship of Magius and Rufinus (or, less frequently, year 950 ''Ab urbe condita''). The denomination 197 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Europe for naming years. Events By place Roman Empire * February 19 – Battle of Lugdunum: Emperor Septimius Severus defeats the self-proclaimed emperor Clodius Albinus at Lugdunum (modern Lyon). Albinus commits suicide; legionaries sack the town. * Septimius Severus returns to Rome and has about 30 of Albinus's supporters in the Senate executed. After his victory he declares himself the adopted son of the late Marcus Aurelius. * Septimius Severus forms new naval units, manning all the triremes in Italy with heavily armed troops for war in the East. His soldiers embark on an artificial canal between the Tigris ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
