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Automorphic Number
In mathematics, an automorphic number (sometimes referred to as a circular number) is a natural number in a given number base b whose square "ends" in the same digits as the number itself. Definition and properties Given a number base b, a natural number n with k digits is an automorphic number if n is a fixed point of the polynomial function f(x) = x^2 over \mathbb/b^k\mathbb, the ring of integers modulo b^k. As the inverse limit of \mathbb/b^k\mathbb is \mathbb_b, the ring of b-adic integers, automorphic numbers are used to find the numerical representations of the fixed points of f(x) = x^2 over \mathbb_b. For example, with b = 10, there are four 10-adic fixed points of f(x) = x^2, the last 10 digits of which are: : \ldots 0000000000 : \ldots 0000000001 : \ldots 8212890625 : \ldots 1787109376 Thus, the automorphic numbers in base 10 are 0, 1, 5, 6, 25, 76, 376, 625, 9376, 90625, 109376, 890625, 2890625, 7109376, 12890625, 87109376, 212890625, 787109376, 1787109376, 82 ... [...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] |
Hensel's Lemma
In mathematics, Hensel's lemma, also known as Hensel's lifting lemma, named after Kurt Hensel, is a result in modular arithmetic, stating that if a univariate polynomial has a simple root modulo a prime number , then this root can be ''lifted'' to a unique root modulo any higher power of . More generally, if a polynomial factors modulo into two coprime polynomials, this factorization can be lifted to a factorization modulo any higher power of (the case of roots corresponds to the case of degree for one of the factors). By passing to the "limit" (in fact this is an inverse limit) when the power of tends to infinity, it follows that a root or a factorization modulo can be lifted to a root or a factorization over the p-adic integer, -adic integers. These results have been widely generalized, under the same name, to the case of polynomials over an arbitrary commutative ring, where is replaced by an ideal (ring theory), ideal, and "coprime polynomials" means "polynomials that gene ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory can often be understood through the study of Complex analysis, analytical objects, such as the Riemann zeta function, that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, as for instance how irrational numbers can be approximated by fractions (Diophantine approximation). Number theory is one of the oldest branches of mathematics alongside geometry. One quirk of number theory is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (mathematics), series, and analytic functions. These theories are usually studied in the context of Real number, real and Complex number, complex numbers and Function (mathematics), functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any Space (mathematics), space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space). History Ancient Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zero-divisor
In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zero divisor if there exists a nonzero in such that . This is a partial case of divisibility in rings. An element that is a left or a right zero divisor is simply called a zero divisor. An element that is both a left and a right zero divisor is called a two-sided zero divisor (the nonzero such that may be different from the nonzero such that ). If the ring is commutative, then the left and right zero divisors are the same. An element of a ring that is not a left zero divisor (respectively, not a right zero divisor) is called left regular or left cancellable (respectively, right regular or right cancellable). An element of a ring that is left and right cancellable, and is hence not a zero divisor, is called regular or cancellable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-adic Analysis
In mathematics, ''p''-adic analysis is a branch of number theory that studies functions of ''p''-adic numbers. Along with the more classical fields of real and complex analysis, which deal, respectively, with functions on the real and complex numbers, it belongs to the discipline of mathematical analysis. The theory of complex-valued numerical functions on the ''p''-adic numbers is part of the theory of locally compact groups ( abstract harmonic analysis). The usual meaning taken for ''p''-adic analysis is the theory of ''p''-adic-valued functions on spaces of interest. Applications of ''p''-adic analysis have mainly been in number theory, where it has a significant role in diophantine geometry and diophantine approximation. Some applications have required the development of ''p''-adic functional analysis and spectral theory. In many ways ''p''-adic analysis is less subtle than classical analysis, since the ultrametric inequality means, for example, that convergence of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-adic Number
In number theory, given a prime number , the -adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; -adic numbers can be written in a form similar to (possibly infinite) decimals, but with digits based on a prime number rather than ten, and extending to the left rather than to the right. For example, comparing the expansion of the rational number \tfrac15 in base vs. the -adic expansion, \begin \tfrac15 &= 0.01210121\ldots \ (\text 3) &&= 0\cdot 3^0 + 0\cdot 3^ + 1\cdot 3^ + 2\cdot 3^ + \cdots \\ mu\tfrac15 &= \dots 121012102 \ \ (\text) &&= \cdots + 2\cdot 3^3 + 1 \cdot 3^2 + 0\cdot3^1 + 2 \cdot 3^0. \end Formally, given a prime number , a -adic number can be defined as a series s=\sum_^\infty a_i p^i = a_k p^k + a_ p^ + a_ p^ + \cdots where is an integer (possibly negative), and each a_i is an integer such that 0\le a_i < p. A -adic integer is a -adic number such that < ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kaprekar Number
In mathematics, a natural number in a given number base is a p-Kaprekar number if the representation of its square in that base can be split into two parts, where the second part has p digits, that add up to the original number. For example, in base 10, 45 is a 2-Kaprekar number, because 45² = 2025, and 20 + 25 = 45. The numbers are named after D. R. Kaprekar. Definition and properties Let n be a natural number. Then the Kaprekar function for base b > 1 and power p > 0 F_ : \mathbb \rightarrow \mathbb is defined to be the following: :F_(n) = \alpha + \beta, where \beta = n^2 \bmod b^p and :\alpha = \frac A natural number n is a p-Kaprekar number if it is a fixed point for F_, which occurs if F_(n) = n. 0 and 1 are trivial Kaprekar numbers for all b and p, all other Kaprekar numbers are nontrivial Kaprekar numbers. The earlier example of 45 satisfies this definition with b = 10 and p = 2, because : \beta = n^2 \bmod b^p = 45^2 \bmod 10^2 = 25 : \alpha = \frac = \frac = 20 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Dynamics
Arithmetic dynamics is a field that amalgamates two areas of mathematics, dynamical systems and number theory. Part of the inspiration comes from complex dynamics, the study of the Iterated function, iteration of self-maps of the complex plane or other complex algebraic varieties. Arithmetic dynamics is the study of the number-theoretic properties of integer point, integer, rational point, rational, p-adic number, -adic, or algebraic points under repeated application of a polynomial or rational function. A fundamental goal is to describe arithmetic properties in terms of underlying geometric structures. ''Global arithmetic dynamics'' is the study of analogues of classical diophantine geometry in the setting of discrete dynamical systems, while ''local arithmetic dynamics'', also called p-adic dynamics, p-adic or nonarchimedean dynamics, is an analogue of complex dynamics in which one replaces the complex numbers by a -adic field such as or and studies chaotic behavior and the Fa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tree Structure
A tree structure, tree diagram, or tree model is a way of representing the hierarchical nature of a structure in a graphical form. It is named a "tree structure" because the classic representation resembles a tree, although the chart is generally upside down compared to a biological tree, with the "stem" at the top and the "leaves" at the bottom. A tree structure is conceptual, and appears in several forms. For a discussion of tree structures in specific fields, see Tree (data structure) for computer science; insofar as it relates to graph theory, see tree (graph theory) or tree (set theory). Other related articles are listed below. Terminology and properties The tree elements are called "nodes". The lines connecting elements are called "branches". Nodes without children are called leaf nodes, "end-nodes", or "leaves". Every finite tree structure has a member that has no superior. This member is called the "root" or root node. The root is the starting node. But the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degree Of A Polynomial
In mathematics, the degree of a polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients. The degree of a term is the sum of the exponents of the variables that appear in it, and thus is a non-negative integer. For a univariate polynomial, the degree of the polynomial is simply the highest exponent occurring in the polynomial. The term order has been used as a synonym of ''degree'' but, nowadays, may refer to several other concepts (see Order of a polynomial (other)). For example, the polynomial 7x^2y^3 + 4x - 9, which can also be written as 7x^2y^3 + 4x^1y^0 - 9x^0y^0, has three terms. The first term has a degree of 5 (the sum of the powers 2 and 3), the second term has a degree of 1, and the last term has a degree of 0. Therefore, the polynomial has a degree of 5, which is the highest degree of any term. To determine the degree of a polynomial that is not in standard form, such as (x+1)^2 - (x-1)^2, one c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
