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The Bernoulli numbers Bn were first discovered in connection with the closed forms of the sums for various fixed values of n. The closed forms are always polynomials in m of degree n + 1 and are called Bernoulli polynomials. The coefficients of the Bernoulli polynomials are closely related to the Bernoulli numbers, as follows: For example, taking n to be 1, we have 0 + 1 + 2 + ... + (m−1) = 1/2 (B0 m2 + 2 B1 m1< ...

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Bernoulli number, Bernoulli number - Introduction, Bernoulli number - Assorted identities, Bernoulli number - Arithmetical properties of the Bernoulli numbers, Bernoulli number - p-adic continuity, Bernoulli number - Geometrical properties of the Bernoulli numbers, Bernoulli number - Efficient computation of Bernoulli numbers mod p

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The Padovan sequence also satisfies the recurrence relations P(n) = P(n − 1) + P(n − 5) P(n) = P(n − 2) + P(n − 4) + P(n − 8) P(n) = P(n − 3) + P(n − 4) + P(n − 5) P(n) = P(n − 4) + P(n − 5) + P(n − 6) + P( ...

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Padovan sequence, Padovan sequence - Recurrence relations, Padovan sequence - Extension to negative parameters, Padovan sequence - Sums of terms, Padovan sequence - Other identities, Padovan sequence - Binet-like formula, Padovan sequence - Combinatorial interpretations, Padovan sequence - Generating function, Padovan sequence - Generalizations, Padovan sequence - Padovan prime, Padovan sequence - Padovan L-System, Padovan sequence - Padovan Cuboid Spiral

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The Arabic letters of the opening phrase of the Qur'an (the Basmala phrase Bismillah al-Rahman al-Rahim "In the name of God, the Compassionate and Merciful") sum to the numerical value 786 in the system of Abjad numerals. Not all Muslims place emphasis on this numerological analysis, but some Muslims in South Asia use the number 786 as an Islamic symbol, and some Muslim scholars and groups (such as the "Qur'an-only" Submitters) ...

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786 number, 786 number - In mathematics, 786 number - In astronomy, 786 number - In religion, 786 number - External link

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Fermat's little theorem states that all prime numbers have that property. In this sense, Carmichael numbers are similar to prime numbers. They are called pseudoprimes. Carmichael numbers are sometimes also called absolute pseudoprimes. Carmichael numbers are important because they can fool the Fermat primality test, thus they are always fermat liars. Since Carmichael numbers exist, this primality test cannot be relied upon to prove the primality of a number, although it can ...

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Carmichael number, Carmichael number - Overview, Carmichael number - Properties, Carmichael number - Higher-order Carmichael numbers, Carmichael number - Properties, Carmichael number - Layman's overview

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Many operations on a Latin square produce another Latin square (for example, turning it upside down). If we permute the rows, permute the columns, and permute the names of the symbols of a Latin square, we obtain a new Latin square said to be isotopic to the first. Isotopism is an equivalence relation, so the set of all Latin squares is divided into subsets, called isotopy classes, such that two squares in the same clas ...

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Latin square, Latin square - Orthogonal array representation, Latin square - Equivalence classes of Latin squares, Latin square - The number of Latin squares, Latin square - Examples, Latin square - Latin squares and mathematical puzzles

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There are several ways of explaining why the formula given for Cn is correct; that is, why it solves the combinatorial problems listed above. The first proof below uses a generating function, and is not particularly illuminating. The second and third proofs are examples of bijective proofs; they involve literally counting a collection of some kind of object to arrive at the correct formula. Catalan number - First proof: using generating functions. The Catalan numbe ...

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Catalan number, Catalan number - Properties of the Catalan numbers, Catalan number - Applications in combinatorics, Catalan number - Proof of the formula, Catalan number - First proof: using generating functions, Catalan number - Second proof, Catalan number - Third proof, Catalan number - Hankel matrix, Catalan number - History

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The Bell numbers can easily be calculated by creating the so-called Bell triangle, also called Aitken's array or the Peirce triangle: Start with the number one. Put this on a row by itself. Start a new row with the rightmost element from the previous row as the leftmost number Determine the numbers not on the left column by taking the sum of the number to the left and the number above the number to the left (the number diagonally up and left of the number we are calculating) Repeat ...

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Bell number, Bell number - Partitions of a set, Bell number - Another view of Bell numbers, Bell number - Properties of Bell numbers, Bell number - Triangle scheme for calculating Bell numbers

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The connection between this function and prime numbers was already realized by Leonhard Euler: an infinite product extending over all prime numbers p. This is called an Euler product formula and is a consequence of two simple and fundamental results in mathematics; the formula for the geometric series and the fundamental theorem of arithmetic. Riemann zeta function - Proving the Euler product formula. Each factor (for a given prime p) in the product above can be expanded to a geometric series consisting of the reciprocal of See also:

Riemann zeta function, Riemann zeta function - Definition, Riemann zeta function - Values at the integers, Riemann zeta function - Relationship to prime numbers, Riemann zeta function - Proving the Euler product formula, Riemann zeta function - An easier proof for the layperson, Riemann zeta function - The importance of the zeros of ζs, Riemann zeta function - Basic properties, Riemann zeta function - The Riemann zeta function as a Mellin transform, Riemann zeta function - Series expansions, Riemann zeta function - Globally convergent series, Riemann zeta function - Universality, Riemann zeta function - Applications, Riemann zeta function - Generalizations, Riemann zeta function - Zeta functions in fiction

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Historically, the definition of repunits was motivated by recreational mathematicians looking for prime factors of such numbers. Wikipedia contains a list of repunit factorizations. It is easy to show that if n is divisible by a, then Rn is divisible by Ra. For example, 9 is divisible by 3, and indeed R9 is divisible by R3—in fact, 111111111 = 111 · 1001001. Thus, for Rn to be prime nSee also:

Repunit, Repunit - Definition, Repunit - Repunit primes, Repunit - Generalizations

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The exponential function ex is important because it is the unique function (up to multiplication by a constant) which is its own derivative, and therefore, its own primitive: and , where C is a constant. It is known that e is both irrational (proof) and transcendental (proof). It was the first number to be proved transcendental without having been specifically constructed for this purpose (cf. Liouville number); the proof was given by Charles Hermite in 1873. It is conjectured to be normal. It features in Euler's formula, one of the most imp ...

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E mathematical constant, E mathematical constant - Definitions, E mathematical constant - Properties, E mathematical constant - History, E mathematical constant - Non-mathematical uses of e, E mathematical constant - Notes

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Here is a table that shows how the three functions π(x), x / ln x and Li(x) compare: x π(x) π(x) − x / ln x Li(x) − π(x) x / π(x) 10 4 −0.3 2.2 2.500 102 25 3.3 5.1 4.000 103 ...

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Prime number theorem, Prime number theorem - Statement of the theorem, Prime number theorem - The prime counting function in terms of the logarithmic integral, Prime number theorem - The issue of depth, Prime number theorem - The prime number theorem for arithmetic progressions, Prime number theorem - Bounds on the prime counting function, Prime number theorem - Approximations for the nth prime number, Prime number theorem - Gaps between primes, Prime number theorem - Table of πx x / ln x and Lix, Prime number theorem - Analogue for irreducible polynomials over a finite field

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The first seven Proth numbers are (sequence A080075 in OEIS): P0 = 21 + 1 = 3 P1 = 22 + 1 = 5 P2 = 23 + 1 = 9 P3 = 3 × 22 + 1 = 13 P4 = 24 + 1 = 17 P5 = 3 × 23 + 1 = 25 See also:

Proth's theorem, Proth's theorem - Numerical examples, Proth's theorem - History

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The molecular structure of the alkanes directly affects their physical and chemical characteristics. It is derived from the electron configuration of carbon, which has four valence electrons. The carbon atoms in alkanes are always sp3-hybridised, that is to say that the valence electrons are said to be in four equivalent orbitals derived from the combination of the 2s-orbital and the three 2p-orbitals. These orbitals, which have identical energies, are arranged spatially in the form of a tetrahedron, the angle of 109.47° between them. ...

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Alkane, Alkane - Isomerism, Alkane - Nomenclature of alkanes, Alkane - Alkanes with unbranched carbon chains, Alkane - Alkanes with branched carbon chains, Alkane - Trivial names, Alkane - Occurrence, Alkane - Purification and use, Alkane - Preparation, Alkane - Molecular geometry, Alkane - Bond lengths and bond angles, Alkane - Conformation, Alkane - Properties, Alkane - Physical properties, Alkane - Chemical properties, Alkane - Thermochemistry, Alkane - Spectroscopic properties, Alkane - Reactions, Alkane - Reactions with oxygen, Alkane - Reactions with halogens, Alkane - Cracking and reforming, Alkane - Other reactions, Alkane - Hazards, Alkane - Alkanes in nature, Alkane - Bacteria and archaea, Alkane - Fungi and plants, Alkane - Animals, Alkane - Ecological relations

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The strategy for solving a puzzle may be regarded as comprising a combination of three processes: scanning, marking up, and analysing. Sudoku - Scanning. Scanning is performed at the outset and periodically throughout the solution. Scans may have to be performed several times in between analysis periods. Scanning consists of two basic techniques: Cross-hatching: the scanning of rows (or columns) to identify which line in a particular region may contain a certain number by a process of e ...

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Sudoku, Sudoku - Introduction, Sudoku - Rules and terminology, Sudoku - Solution methods, Sudoku - Scanning, Sudoku - Marking up, Sudoku - Analysis, Sudoku - Computer solutions, Sudoku - Difficulty ratings, Sudoku - Construction, Sudoku - Variants, Sudoku - Mathematics of Sudoku, Sudoku - History, Sudoku - Popularity in the media

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By defining the quantities a series of relationships can be given in the form where An,Bn,Cn and Dn are conjectured to be positive integers. Plouffe gives a table of values: A recurrence relatio ...

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Zeta constants, Zeta constants - ζ3, Zeta constants - ζ5, Zeta constants - ζ7, Zeta constants - ζ2n+1, Zeta constants - ζ2n

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N serves as an alveolar nasal in virtually all languages that use the Latin alphabet. A common digraph with N is NG, which produces a velar nasal in a variety of languages, usually final in English. Aspirated forms NH and NGH are sometimes seen in other languages. In the International Phonetic Alphabet, the lowercase [n] represents the alveolar nasal sound. A small capital [ɴ] represents the uvular nasal. ...

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N, N - Usage, N - Alternative representations, N - Computing, N - Meanings for N

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Pi - Geometry. π appears in many formulae in geometry involving circles and spheres. (All of these are a consequence of the first one, as the area of a circle can be written as A = ∫(2πr)dr ("sum of annuli of infinitesimal width"), and others concern a surface or solid of revolution.) Also, the angle measure of 180° (degrees) is equal to π radians. ...

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Pi, Pi - Properties, Pi - Formulae involving π, Pi - Geometry, Pi - Analysis, Pi - Continued fractions, Pi - Number theory, Pi - Dynamical systems and ergodic theory, Pi - Physics, Pi - Probability and statistics, Pi - History of π, Pi - Numerical approximations of π, Pi - Miscellaneous formulae, Pi - Less accurate approximations, Pi - Open questions, Pi - The nature of π, Pi - Fictional references, Pi - π culture

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All superior highly composite numbers are highly composite; it can also be shown that there exist prime numbers π1, π2, ... such that the n-th superior highly composite number sn can be written as The first few πn are 2, 3, 2, 5, 2, 3, 7, ... (sequence A000705 in OEIS). ...

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Superior highly composite number, Superior highly composite number - Properties, Superior highly composite number - External links

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Forty-three is the 14th smallest prime number. The previous is forty-one, with which it comprises a twin prime, the next is forty-seven. 43 is a centered heptagonal number. 43 is the smallest prime that is not a Chen prime. Let a(0) = a(1) = 1, and thenceforth a(n) = (a(0)2 + a(1)2 + ... + a(n-1)2) / (n-1). This sequence continues 1 1 2 3 5 10 28 154... (sequence A003504 in OEIS). Amazingly, a(43) is the first term of this sequence that is not an integer. Negative forty ...

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43 number, 43 number - In mathematics, 43 number - In science, 43 number - In the periodic table, 43 number - In astronomy, 43 number - In other fields

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Three hundred is In bowling, a perfect score, achieved by rolling strikes in all ten frames. The title of a comic book by Frank Miller about the Battle of Thermopylae. The lowest possible credit score. For the year, see 300. ...

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300 number, 300 number - Mathematical properties, 300 number - Other fields, 300 number - Integers from 301 to 399

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Finding all solutions to the eight queens puzzle is a good example of a simple but nontrivial problem. For this reason, it is often used as an example problem for various programming techniques, including nontraditional approaches such as constraint programming, logic programming or genetic algorithms. Most often, it is used as an example of a problem which can be solved with a recursive algorithm, by phrasing the n queens problem inductively in terms of adding a single queen to any solution to the n−1 queens problem. ...

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Eight queens puzzle, Eight queens puzzle - History, Eight queens puzzle - Constructing a solution, Eight queens puzzle - Counting all solutions, Eight queens puzzle - Related problems, Eight queens puzzle - The eight queens puzzle as an exercise in algorithm design, Eight queens puzzle - A standard recursive solution, Eight queens puzzle - A constraint logic programming solution, Eight queens puzzle - An iterative solution

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In the light of later work on Galois theory, the principles of these proofs have been clarified. It is straightforward to show from analytic geometry that constructible lengths must come from base lengths by the solution of some sequence of quadratic equations. In terms of field theory, such lengths must be contained in a field extension generated by a tower of quadratic extensions. It follows that a field generated by constructions will always have degree over the base field that is a power of two. In the specific case of a regular n-gon, the question reduces to the question of cons ...

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Constructible polygon, Constructible polygon - Conditions for constructibility, Constructible polygon - General theory, Constructible polygon - Detailed results in terms of Fermat primes, Constructible polygon - Compass-and-straightedge constructions, Constructible polygon - Other constructions

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