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Sums Of Powers
In mathematics and statistics, sums of powers occur in a number of contexts: * Sums of squares arise in many contexts. For example, in geometry, the Pythagorean theorem involves the sum of two squares; in number theory, there are Legendre's three-square theorem and Jacobi's four-square theorem; and in statistics, the analysis of variance involves summing the squares of quantities. *There are only finitely many positive integers that are not sums of ''distinct'' squares. The largest one is 128. The same applies for sums of distinct cubes (largest one is 12,758), distinct fourth powers (largest is 5,134,240), etc. See for a generalization to sums of polynomials. *Faulhaber's formula expresses 1^k + 2^k + 3^k + \cdots + n^k as a polynomial in , or alternatively in terms of a Bernoulli polynomial. *Fermat's right triangle theorem states that there is no solution in positive integers for a^2=b^4+c^4 and a^4=b^4+c^2. *Fermat's Last Theorem states that x^k+y^k=z^k is impossible in posit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beal's Conjecture
The Beal conjecture is the following conjecture in number theory: :If :: A^x +B^y = C^z, :where ''A'', ''B'', ''C'', ''x'', ''y'', and ''z'' are positive integers with ''x'', ''y'', ''z'' > 2, then ''A'', ''B'', and ''C'' have a common prime factor. Equivalently, :The equation A^x + B^y = C^z has no solutions in positive integers and pairwise coprime integers ''A, B, C'' if ''x, y, z'' > 2. The conjecture was formulated in 1993 by Andrew Beal, a banker and amateur mathematician, while investigating generalizations of Fermat's Last Theorem. Since 1997, Beal has offered a monetary prize for a peer-reviewed proof of this conjecture or a counterexample. The value of the prize has increased several times and is currently $1 million. In some publications, this conjecture has occasionally been referred to as a generalized Fermat equation, the Mauldin conjecture, and the Tijdeman-Zagier conjecture. Related examples To illustrate, the solution 3^3 + 6^3 = 3^5 has bases with a commo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Sum Symmetric Polynomial
In mathematics, specifically in commutative algebra, the power sum symmetric polynomials are a type of basic building block for symmetric polynomials, in the sense that every symmetric polynomial with rational coefficients can be expressed as a sum and difference of products of power sum symmetric polynomials with rational coefficients. However, not every symmetric polynomial with integral coefficients is generated by integral combinations of products of power-sum polynomials: they are a generating set over the ''rationals,'' but not over the ''integers.'' Definition The power sum symmetric polynomial of degree ''k'' in n variables ''x''1, ..., ''x''''n'', written ''p''''k'' for ''k'' = 0, 1, 2, ..., is the sum of all ''k''th powers of the variables. Formally, : p_k (x_1, x_2, \dots,x_n) = \sum_^n x_i^k \, . The first few of these polynomials are :p_0 (x_1, x_2, \dots,x_n) = 1 + 1 + \cdots + 1 = n \, , :p_1 (x_1, x_2, \dots,x_n) = x_1 + x_2 + \cdots + x_n \, , :p_2 (x_1, x_2, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermat Cubic
In geometry, the Fermat cubic, named after Pierre de Fermat, is a surface defined by : x^3 + y^3 + z^3 = 1. \ Methods of algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ... provide the following parameterization of Fermat's cubic: : x(s,t) = : y(s,t) = : z(s,t) = . In projective space the Fermat cubic is given by :w^3+x^3+y^3+z^3=0. The 27 lines lying on the Fermat cubic are easy to describe explicitly: they are the 9 lines of the form (''w'' : ''aw'' : ''y'' : ''by'') where ''a'' and ''b'' are fixed numbers with cube −1, and their 18 conjugates under permutations of coordinates. ::::''Real points of Fermat cubic surface.'' References * * Algebraic surfaces {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cube (algebra)
In arithmetic and algebra, the cube of a number is its third exponentiation, power, that is, the result of multiplying three instances of together. The cube of a number is denoted , using a superscript 3, for example . The cube Mathematical operation, operation can also be defined for any other expression (mathematics), mathematical expression, for example . The cube is also the number multiplied by its square (algebra), square: :. The ''cube function'' is the function (mathematics), function (often denoted ) that maps a number to its cube. It is an odd function, as :. The volume of a Cube (geometry), geometric cube is the cube of its side length, giving rise to the name. The Inverse function, inverse operation that consists of finding a number whose cube is is called extracting the cube root of . It determines the side of the cube of a given volume. It is also raised to the one-third power. The graph of a function, graph of the cube function is known as the cubic para ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Newton's Identities
In mathematics, Newton's identities, also known as the Girard–Newton formulae, give relations between two types of symmetric polynomials, namely between power sums and elementary symmetric polynomials. Evaluated at the roots of a monic polynomial ''P'' in one variable, they allow expressing the sums of the ''k''-th powers of all roots of ''P'' (counted with their multiplicity) in terms of the coefficients of ''P'', without actually finding those roots. These identities were found by Isaac Newton around 1666, apparently in ignorance of earlier work (1629) by Albert Girard. They have applications in many areas of mathematics, including Galois theory, invariant theory, group theory, combinatorics, as well as further applications outside mathematics, including general relativity. Mathematical statement Formulation in terms of symmetric polynomials Let ''x''1, ..., ''x''''n'' be variables, denote for ''k'' ≥ 1 by ''p''''k''(''x''1, ..., ''x''''n'') the ''k''-th p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Golden Ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if \frac = \frac = \varphi, where the Greek letter Phi (letter), phi ( or ) denotes the golden ratio. The constant satisfies the quadratic equation and is an irrational number with a value of The golden ratio was called the extreme and mean ratio by Euclid, and the divine proportion by Luca Pacioli; it also goes by other names. Mathematicians have studied the golden ratio's properties since antiquity. It is the ratio of a regular pentagon's diagonal to its side and thus appears in the Straightedge and compass construction, construction of the dodecahedron and icosahedron. A golden rectangle—that is, a rectangle with an aspect ratio of —may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has bee ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Waring's Problem
In number theory, Waring's problem asks whether each natural number ''k'' has an associated positive integer ''s'' such that every natural number is the sum of at most ''s'' natural numbers raised to the power ''k''. For example, every natural number is the sum of at most 4 squares, 9 cubes, or 19 fourth powers. Waring's problem was proposed in 1770 by Edward Waring, after whom it is named. Its affirmative answer, known as the Hilbert–Waring theorem, was provided by Hilbert in 1909. Waring's problem has its own Mathematics Subject Classification, 11P05, "Waring's problem and variants". Relationship with Lagrange's four-square theorem Long before Waring posed his problem, Diophantus had asked whether every positive integer could be represented as the sum of four perfect squares greater than or equal to zero. This question later became known as Bachet's conjecture, after the 1621 translation of Diophantus by Claude Gaspard Bachet de Méziriac, and it was solved by Joseph-Louis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature, "imaginary" complex numbers have a mathematical existence as firm as that of the real numbers, and they are fundamental tools in the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sum Of Reciprocals
In mathematics and especially number theory, the sum of reciprocals (or sum of inverses) generally is computed for the reciprocals of some or all of the positive integers (counting numbers)—that is, it is generally the sum of unit fractions. If infinitely many numbers have their reciprocals summed, generally the terms are given in a certain sequence and the first ''n'' of them are summed, then one more is included to give the sum of the first ''n''+1 of them, etc. If only finitely many numbers are included, the key issue is usually to find a simple expression for the value of the sum, or to require the sum to be less than a certain value, or to determine whether the sum is ever an integer. For an infinite series of reciprocals, the issues are twofold: First, does the sequence of sums diverge—that is, does it eventually exceed any given number—or does it converge, meaning there is some number that it gets arbitrarily close to without ever exceeding it? (A set of positiv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Zeta Function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for and its analytic continuation elsewhere. The Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics. Leonhard Euler first introduced and studied the function over the reals in the first half of the eighteenth century. Bernhard Riemann's 1859 article "On the Number of Primes Less Than a Given Magnitude" extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation, and established a relation between its zeros and the distribution of prime numbers. This paper also contained the Riemann hypothesis, a conjecture about the distribution of complex zeros of the Riemann zeta function that many mathematicians consider th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
