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Constructible Polygon
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., a constructible polygon is a regular polygon In , a regular polygon is a that is (all angles are equal in measure) and (all sides have the same length). Regular polygons may be either or . In the , a sequence of regular polygons with an increasing number of sides approximates a , if the ... that can be constructed with compass and straightedge. For example, a regular pentagon In geometry, a pentagon (from the Greek language, Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is ... is constructible with compass and straightedge while a regular heptagon In geometry, ...
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Pentagon Construct
In geometry, a pentagon (from the Greek language, Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°. A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ''regular pentagon'' (or ''star polygon, star pentagon'') is called a pentagram. Regular pentagons A ''regular polygon, regular pentagon'' has Schläfli symbol and interior angles of 108°. A ''regular polygon, regular pentagon'' has five lines of reflectional symmetry, and rotational symmetry of order 5 (through 72°, 144°, 216° and 288°). The diagonals of a convex polygon, convex regular pentagon are in the golden ratio to its sides. Its height (distance from one side to the opposite vertex) and width (distance between two farthest separated points, which equals the diagonal length) are given by :\text = \frac \cdot \t ...
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Pierre Wantzel
Pierre Laurent Wantzel (5 June 1814 in Paris – 21 May 1848 in Paris) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ... who proved that several ancient geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that are related with distance, shape, size, ... problems were impossible to solve using only compass and straightedge Straightedge and compass construction, also known as ruler-and-compass construction or classical construction, is the construction of lengths, angle In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandr .... In a paper from 1837, ...
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Fermat Primes
In mathematics, a Fermat number, named after Pierre de Fermat, who first studied them, is a natural number, positive integer of the form :F_ = 2^ + 1, where ''n'' is a non-negative integer. The first few Fermat numbers are: : 3 (number), 3, 5 (number), 5, 17 (number), 17, 257 (number), 257, 65537 (number), 65537, 4294967297, 18446744073709551617, ... . If 2''k'' + 1 is Prime number, prime and ''k'' > 0, then ''k'' must be a power of 2, so 2''k'' + 1 is a Fermat number; such primes are called Fermat primes. As of 2021, the only known Fermat primes are ''F''0 = 3, ''F''1 = 5, ''F''2 = 17, ''F''3 = 257, and ''F''4 = 65537 ; heuristics suggest that there are no more. Basic properties The Fermat numbers satisfy the following recurrence relations: : F_ = (F_-1)^+1 : F_ = F_ \cdots F_ + 2 for ''n'' ≥ 1, : F_ = F_ + 2^F_ \cdots F_ : F_ = F_^2 - 2(F_-1)^2 for ''n'' ≥ 2. Each of these relations can be proved by mathematical induction. From the second equation, we can deduce Gol ...
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Integer
An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to be in relation with") is "an appa ... ''integer'' meaning "whole") is colloquially defined as a number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduct ... that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and  are not. The set of integers consists of zero (), the positive natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' larges ...
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Cyclotomic Polynomial
In mathematics, the ''n''th cyclotomic polynomial, for any positive integer ''n'', is the unique irreducible polynomial with integer coefficients that is a divisor of x^n-1 and is not a divisor of x^k-1 for any Its root of a function, roots are all ''n''th primitive root of unity, primitive roots of unity e^ , where ''k'' runs over the positive integers not greater than ''n'' and coprime integers, coprime to ''n'' (and ''i'' is the imaginary unit). In other words, the ''n''th cyclotomic polynomial is equal to : \Phi_n(x) = \prod_\stackrel \left(x-e^\right). It may also be defined as the monic polynomial with integer coefficients that is the minimal polynomial (field theory), minimal polynomial over the Field (mathematics), field of the rational numbers of any Root of unity#Definition, primitive ''n''th-root of unity ( e^ is an example of such a root). An important relation linking cyclotomic polynomials and primitive roots of unity is :\prod_\Phi_d(x) = x^n - 1, showing that ...
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Root Of A Function
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., a zero (also sometimes called a root) of a real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...-, complex The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London , mottoeng = Let all come who by merit deserve the most reward , established = , type = Public university, Public rese ...-, or generally vector-valued function A vector-valued function, also referred to as a vector function, is a function (mathematics), mathematical function ...
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Square Root
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., a square root of a number is a number such that ; in other words, a number whose ''square In Euclidean geometry, a square is a regular The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * Regular (Badfinger ...'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . Every nonnegative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical ...
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Constructible Number
In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ... and algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ..., a real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ... r is constructible if and only if, given a line segment of unit length, a line segment of length , r, can be constructed with compass and straightedge Straightedge and compass constructi ...
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Number Theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study s as well as the properties of mathematical objects made out of integers (for example, s) or defined as generalizations of the integers (for example, s). Integers can be considered either in themselves or as solutions to equations (). Questions in number theory are often best understood through the study of objects (for example, the ) that encode properties of the integers, primes or other number-theoretic objects in some fashion (). One may also study s in relation to rational numbers, for example, as approximated by the latter (). The older term for number theory is ''arithmet ...
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Geometry
Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cra ..., one of the oldest branches of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal .... It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a geometer A geometer is a mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancie ...
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Prime Number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), 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 () 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 factorization, 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 t ...
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