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Effective Results In Number Theory
For historical reasons and in order to have application to the solution of Diophantine equations, results in number theory have been scrutinised more than in other branches of mathematics to see if their content is effectively computable. Where it is asserted that some list of integers is finite, the question is whether in principle the list could be printed out after a machine computation. Littlewood's result An early example of an ineffective result was J. E. Littlewood's theorem of 1914, that in the prime number theorem the differences of both ψ(''x'') and π(''x'') with their asymptotic estimates change sign infinitely often. In 1933 Stanley Skewes obtained an effective upper bound for the first sign change, now known as Skewes' number. In more detail, writing for a numerical sequence ''f'' (''n''), an ''effective'' result about its changing sign infinitely often would be a theorem including, for every value of ''N'', a value ''M'' > ''N'' such that ''f'' ('' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diophantine Equation
In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a constant the sum of two or more monomials, each of degree one. An exponential Diophantine equation is one in which unknowns can appear in exponents. Diophantine problems have fewer equations than unknowns and involve finding integers that solve simultaneously all equations. As such systems of equations define algebraic curves, algebraic surfaces, or, more generally, algebraic sets, their study is a part of algebraic geometry that is called ''Diophantine geometry''. The word ''Diophantine'' refers to the Hellenistic mathematician of the 3rd century, Diophantus of Alexandria, who made a study of such equations and was one of the first mathematicians to introduce symbolism into algebra. The mathematical study of Diophantine problems that Di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thue–Siegel–Roth Theorem
In mathematics, Roth's theorem is a fundamental result in diophantine approximation to algebraic numbers. It is of a qualitative type, stating that algebraic numbers cannot have many rational number approximations that are 'very good'. Over half a century, the meaning of ''very good'' here was refined by a number of mathematicians, starting with Joseph Liouville in 1844 and continuing with work of , , , and . Statement Roth's theorem states that every irrational algebraic number \alpha has approximation exponent equal to 2. This means that, for every \varepsilon>0, the inequality :\left, \alpha - \frac\ \frac with C(\alpha,\varepsilon) a positive number depending only on \varepsilon>0 and \alpha. Discussion The first result in this direction is Liouville's theorem on approximation of algebraic numbers, which gives an approximation exponent of ''d'' for an algebraic number α of degree ''d'' ≥ 2. This is already enough to demonstrate the existence of transcen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Denominator
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A ''common'', ''vulgar'', or ''simple'' fraction (examples: \tfrac and \tfrac) consists of a numerator, displayed above a line (or before a slash like ), and a non-zero denominator, displayed below (or after) that line. Numerators and denominators are also used in fractions that are not ''common'', including compound fractions, complex fractions, and mixed numerals. In positive common fractions, the numerator and denominator are natural numbers. The numerator represents a number of equal parts, and the denominator indicates how many of those parts make up a unit or a whole. The denominator cannot be zero, because zero parts can never make up a whole. For example, in the fraction , the numerator 3 indicates that the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Number
An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the polynomial . That is, it is a value for x for which the polynomial evaluates to zero. As another example, the complex number 1 + i is algebraic because it is a root of . All integers and rational numbers are algebraic, as are all roots of integers. Real and complex numbers that are not algebraic, such as and , are called transcendental numbers. The set of algebraic numbers is countably infinite and has measure zero in the Lebesgue measure as a subset of the uncountable complex numbers. In that sense, almost all complex numbers are transcendental. Examples * All rational numbers are algebraic. Any rational number, expressed as the quotient of an integer and a (non-zero) natural number , satisfies the above definition, because is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rational numbers, also referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by boldface , or blackboard bold \mathbb. A rational number is a real number. The real numbers that are rational are those whose decimal expansion either terminates after a finite number of digits (example: ), or eventually begins to repeat the same finite sequence of digits over and over (example: ). This statement is true not only in base 10, but also in every other integer base, such as the binary and hexadecimal ones (see ). A real number that is not rational is called irrational. Irrational numbers include , , , and . Since the set of rational numbers is countable, and the set of real numbers is uncountable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a field that contains \mathbb and has finite dimension when considered as a vector space over The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory. This study reveals hidden structures behind usual rational numbers, by using algebraic methods. Definition Prerequisites The notion of algebraic number field relies on the concept of a field. A field consists of a set of elements together with two operations, namely addition, and multiplication, and some distributivity assumptions. A prominent example of a field is the field of rational numbers, commonly denoted together with its usual operations of addition and multiplication. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ideal Class Group
In number theory, the ideal class group (or class group) of an algebraic number field is the quotient group where is the group of fractional ideals of the ring of integers of , and is its subgroup of principal ideals. The class group is a measure of the extent to which unique factorization fails in the ring of integers of . The order of the group, which is finite, is called the class number of . The theory extends to Dedekind domains and their field of fractions, for which the multiplicative properties are intimately tied to the structure of the class group. For example, the class group of a Dedekind domain is trivial if and only if the ring is a unique factorization domain. History and origin of the ideal class group Ideal class groups (or, rather, what were effectively ideal class groups) were studied some time before the idea of an ideal was formulated. These groups appeared in the theory of quadratic forms: in the case of binary integral quadratic forms, as put into s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Class Number (number Theory)
In number theory, the ideal class group (or class group) of an algebraic number field is the quotient group where is the group of fractional ideals of the ring of integers of , and is its subgroup of principal ideals. The class group is a measure of the extent to which unique factorization fails in the ring of integers of . The order of the group, which is finite, is called the class number of . The theory extends to Dedekind domains and their field of fractions, for which the multiplicative properties are intimately tied to the structure of the class group. For example, the class group of a Dedekind domain is trivial if and only if the ring is a unique factorization domain. History and origin of the ideal class group Ideal class groups (or, rather, what were effectively ideal class groups) were studied some time before the idea of an ideal was formulated. These groups appeared in the theory of quadratic forms: in the case of binary integral quadratic forms, as put into s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Siegel–Walfisz Theorem
In analytic number theory, the Siegel–Walfisz theorem was obtained by Arnold Walfisz as an application of a theorem by Carl Ludwig Siegel to primes in arithmetic progressions. It is a refinement both of the prime number theorem and of Dirichlet's theorem on primes in arithmetic progressions. Statement Define :\psi(x;q,a) = \sum_\Lambda(n), where \Lambda denotes the von Mangoldt function, and let ''φ'' denote Euler's totient function. Then the theorem states that given any real number ''N'' there exists a positive constant ''C''''N'' depending only on ''N'' such that :\psi(x;q,a)=\frac+O\left(x\exp\left(-C_N(\log x)^\frac\right)\right), whenever (''a'', ''q'') = 1 and :q\le(\log x)^N. Remarks The constant ''C''''N'' is not effectively computable because Siegel's theorem is ineffective. From the theorem we can deduce the following bound regarding the prime number theorem for arithmetic progressions: If, for (''a'', ''q'') = 1, by \pi(x;q,a) we denote the number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Siegel Zero
Siegel (also Segal or Segel), is a German and Ashkenazi Jewish surname. it can be traced to 11th century Bavaria and was used by people who made wax seals for or sealed official documents (each such male being described as a ''Siegelbeamter''). Alternate spellings include Sigel, Sigl, Siegl, and others. "Siegel" is also the modern German word for seal. The name ultimately derives from the Latin ''sigillum,'' meaning "seal" as in the Seal of the City of New York: ''Sigillum Civitatis Novi Eboraci''. The Germanicized derivative of the name was given to professional seal makers and engravers. Some researchers have attributed the surname to Sigel, referring to Sól (Sun), the goddess of the sun in Germanic mythology (Siȝel or sigel in Old English / Anglo-Saxon), but that is highly speculative. Variants, and false cognates Other variants may routinely include Siegelman, Siegle, Sigl, and Sigel. Presumably, some bearers of these names are lineal descendants of ethnic Jews who chan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quarterly Journal Of Mathematics
The ''Quarterly Journal of Mathematics'' is a quarterly peer-reviewed mathematics journal established in 1930 from the merger of ''The Quarterly Journal of Pure and Applied Mathematics'' and the ''Messenger of Mathematics''. According to the ''Journal Citation Reports'', the journal has a 2020 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as i ... of 0.681. References External links * {{Official website, http://qjmath.oxfordjournals.org/ Mathematics journals Publications established in 1930 English-language journals Oxford University Press academic journals Quarterly journals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Class Number 1 Problem
In mathematics, the Gauss class number problem (for imaginary quadratic fields), as usually understood, is to provide for each ''n'' ≥ 1 a complete list of imaginary quadratic fields \mathbb(\sqrt) (for negative integers ''d'') having class number ''n''. It is named after Carl Friedrich Gauss. It can also be stated in terms of discriminants. There are related questions for real quadratic fields and for the behavior as d \to -\infty. The difficulty is in effective computation of bounds: for a given discriminant, it is easy to compute the class number, and there are several ineffective lower bounds on class number (meaning that they involve a constant that is not computed), but effective bounds (and explicit proofs of completeness of lists) are harder. Gauss's original conjectures The problems are posed in Gauss's Disquisitiones Arithmeticae of 1801 (Section V, Articles 303 and 304). are a set of more precise conjectures about the structure of class groups of quadratic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
