Sum Of Two Squares Theorem
In number theory, the sum of two squares theorem relates the prime decomposition of any integer to whether it can be written as a sum of two Square number, squares, such that for some integers , . An integer greater than one can be written as a sum of two squares if and only if its prime decomposition contains no factor , where Prime number, prime p \equiv 3 \pmod 4 and is Parity (mathematics), odd. In writing a number as a sum of two squares, it is allowed for one of the squares to be zero, or for both of them to be equal to each other, so all squares and all doubles of squares are included in the numbers that can be represented in this way. This theorem supplements Fermat's theorem on sums of two squares which says when a prime number can be written as a sum of two squares, in that it also covers the case for composite numbers. A number may have multiple representations as a sum of two squares, counted by the sum of squares function; for instance, every Pythagorean triple a^ ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Line Segment
In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special case of an ''arc (geometry), arc'', with zero curvature. The length of a line segment is given by the Euclidean distance between its endpoints. A closed line segment includes both endpoints, while an open line segment excludes both endpoints; a half-open line segment includes exactly one of the endpoints. In geometry, a line segment is often denoted using an overline (vinculum (symbol), vinculum) above the symbols for the two endpoints, such as in . Examples of line segments include the sides of a triangle or square. More generally, when both of the segment's end points are vertices of a polygon or polyhedron, the line segment is either an edge (geometry), edge (of that polygon or polyhedron) if they are adjacent vertices, or a diagonal. Wh ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Additive Number Theory
Additive number theory is the subfield of number theory concerning the study of subsets of integers and their behavior under addition. More abstractly, the field of additive number theory includes the study of abelian groups and commutative semigroups with an operation of addition. Additive number theory has close ties to combinatorial number theory and the geometry of numbers. Principal objects of study include the sumset of two subsets and of elements from an abelian group , :A + B = \, and the -fold sumset of , :hA = \underset\,. Additive number theory The field is principally devoted to consideration of ''direct problems'' over (typically) the integers, that is, determining the structure of from the structure of : for example, determining which elements can be represented as a sum from , where ' is a fixed subset.Nathanson (1996) II:1 Two classical problems of this type are the Goldbach conjecture (which is the conjecture that contains all even numbers greater than two, ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Sum Of Squares Function
In number theory, the sum of squares function is an arithmetic function that gives the number of representations for a given positive integer as the sum of squares, where representations that differ only in the order of the summands or in the signs of the numbers being squared are counted as different. It is denoted by . Definition The function is defined as :r_k(n) = , \, where , \,\ , denotes the cardinality of a set. In other words, is the number of ways can be written as a sum of squares. For example, r_2(1) = 4 since 1 = 0^2 + (\pm 1)^2 = (\pm 1)^2 + 0^2 where each sum has two sign combinations, and also r_2(2) = 4 since 2 = (\pm 1)^2 + (\pm 1)^2 with four sign combinations. On the other hand, r_2(3) = 0 because there is no way to represent 3 as a sum of two squares. Formulae ''k'' = 2 The number of ways to write a natural number as sum of two squares is given by . It is given explicitly by :r_2(n) = 4(d_1(n)-d_3(n)) where is the number of divisors o ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Lagrange's Four-square Theorem
Lagrange's four-square theorem, also known as Bachet's conjecture, states that every natural number, nonnegative integer can be represented as a sum of four non-negative integer square number, squares. That is, the squares form an additive basis of order four: p = a^2 + b^2 + c^2 + d^2, where the four numbers a, b, c, d are integers. For illustration, 3, 31, and 310 can be represented as the sum of four squares as follows: \begin 3 & = 1^2+1^2+1^2+0^2 \\[3pt] 31 & = 5^2+2^2+1^2+1^2 \\[3pt] 310 & = 17^2+4^2+2^2+1^2 \\[3pt] & = 16^2 + 7^2 + 2^2 +1^2 \\[3pt] & = 15^2 + 9^2 + 2^2 +0^2 \\[3pt] & = 12^2 + 11^2 + 6^2 + 3^2. \end This theorem was proven by Joseph Louis Lagrange in 1770. It is a special case of the Fermat polygonal number theorem. Historical development From examples given in the ''Arithmetica,'' it is clear that Diophantus was aware of the theorem. This book was translated in 1621 into Latin by Claude Gaspard Bachet de Méziriac, Bachet (Claude Gaspard Bachet de Mézi ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Legendre's Three-square Theorem
In mathematics, Legendre's three-square theorem states that a natural number can be represented as the sum of three squares of integers :n = x^2 + y^2 + z^2 if and only if is not of the form n = 4^a(8b + 7) for nonnegative integers and . The first numbers that cannot be expressed as the sum of three squares (i.e. numbers that can be expressed as n = 4^a(8b + 7)) are :7, 15, 23, 28, 31, 39, 47, 55, 60, 63, 71 ... . History Pierre de Fermat gave a criterion for numbers of the form 8''a'' + 1 and 8''a'' + 3 to be sums of a square plus twice another square, but did not provide a proof. N. Beguelin noticed in 1774 that every positive integer which is neither of the form 8''n'' + 7, nor of the form 4''n'', is the sum of three squares, but did not provide a satisfactory proof. In 1796 Gauss proved his Eureka theorem that every positive integer ''n'' is the sum of 3 triangular numbers; this is equivalent to the fact that 8''n'' + 3 is a ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Jacobi Triple Product
In mathematics, the Jacobi triple product is the identity: :\prod_^\infty \left( 1 - x^\right) \left( 1 + x^ y^2\right) \left( 1 +\frac\right) = \sum_^\infty x^ y^, for complex numbers ''x'' and ''y'', with , ''x'', < 1 and ''y'' ≠ 0. It was introduced by in his work '' Fundamenta Nova Theoriae Functionum Ellipticarum''. The Jacobi triple product identity is the Macdonald identity for the affine root system of type ''A''1, and is the Weyl denominator formula for the corresponding affine Kac–Moody algebra. Properties Jacobi's proof relies on Euler's pentagonal number theorem, which is itself a specific case of the Jacobi triple product identity. Let x=q\sqrt q and y^2=-\sqrt. Then we have :\phi(q) = \prod_^\infty \left(1-q^m \right) = \sum_^\infty (-1)^n q^. The Rogers–Ramanujan identities follow with x=q^2\sqrt q, y^2=-\sqrt and x=q^2\sqrt q, y^2=-q\sqrt. The Jacobi Triple Product also allows the Jacobi theta function to be written as an infinite product a ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Unique Factorization Domain
In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is an integral domain (a nontrivial commutative ring in which the product of any two non-zero elements is non-zero) in which every non-zero non- unit element can be written as a product of irreducible elements, uniquely up to order and units. Important examples of UFDs are the integers and polynomial rings in one or more variables with coefficients coming from the integers or from a field. Unique factorization domains appear in the following chain of class inclusions: Definition Formally, a unique factorization domain is defined to be an integral domain ''R'' in which every non-zero element ''x'' of ''R'' which is not a unit can be written as a finite product of irreducible elements ''p''''i'' of ''R'': : ''x'' = ''p''1 ''p''2 � ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Elliptic Function
In the mathematical field of complex analysis, elliptic functions are special kinds of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Those integrals are in turn named elliptic because they first were encountered for the calculation of the arc length of an ellipse. Important elliptic functions are Jacobi elliptic functions and the Weierstrass \wp-function. Further development of this theory led to hyperelliptic functions and modular forms. Definition A meromorphic function is called an elliptic function, if there are two \mathbb- linear independent complex numbers \omega_1,\omega_2\in\mathbb such that : f(z + \omega_1) = f(z) and f(z + \omega_2) = f(z), \quad \forall z\in\mathbb. So elliptic functions have two periods and are therefore doubly periodic functions. Period lattice and fundamental domain If f is an elliptic function with periods \omega_1,\omega_2 it also holds ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Quadratic Form
In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example, 4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to a fixed field , such as the real or complex numbers, and one speaks of a quadratic form ''over'' . Over the reals, a quadratic form is said to be '' definite'' if it takes the value zero only when all its variables are simultaneously zero; otherwise it is '' isotropic''. Quadratic forms occupy a central place in various branches of mathematics, including number theory, linear algebra, group theory ( orthogonal groups), differential geometry (the Riemannian metric, the second fundamental form), differential topology ( intersection forms of manifolds, especially four-manifolds), Lie theory (the Killing form), and statistics (where the exponent of a zero-mean multivariate normal distribution has the quadratic form -\mathbf^\math ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Number Of Divisors
In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as ''the'' divisor function, it counts the ''number of divisors of an integer'' (including 1 and the number itself). It appears in a number of remarkable identities, including relationships on the Riemann zeta function and the Eisenstein series of modular forms. Divisor functions were studied by Ramanujan, who gave a number of important congruences and identities; these are treated separately in the article Ramanujan's sum. A related function is the divisor summatory function, which, as the name implies, is a sum over the divisor function. Definition The sum of positive divisors function ''σ''''z''(''n''), for a real or complex number ''z'', is defined as the sum of the ''z''th powers of the positive divisors of ''n''. It can be expressed in sigma notation as :\sigma_z(n)=\sum_ d^z\,\! , where is shorthand for "''d'' ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Brahmagupta–Fibonacci Identity
In algebra, the Brahmagupta–Fibonacci identity expresses the product of two sums of two squares as a sum of two squares in two different ways. Hence the set of all sums of two squares is closed under multiplication. Specifically, the identity says :\begin \left(a^2 + b^2\right)\left(c^2 + d^2\right) & = \left(ac-bd\right)^2 + \left(ad+bc\right)^2 & & (1) \\ & = \left(ac+bd\right)^2 + \left(ad-bc\right)^2. & & (2) \end For example, :(1^2 + 4^2)(2^2 + 7^2) = 26^2 + 15^2 = 30^2 + 1^2. The identity is also known as the Diophantus identity, Daniel Shanks, Solved and unsolved problems in number theory, p.209, American Mathematical Society, Fourth edition 1993. as it was first proved by Diophantus of Alexandria. It is a special case of Euler's four-square identity, and also of Lagrange's identity. Brahmagupta proved and used a more general Brahmagupta identity, stating :\begin \left(a^2 + nb^2\right)\left(c^2 + nd^2\right) & = \left(a ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |