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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] |
Elementary 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 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 that it deals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
A 150x150 Ulam Spiral Of Dots With Varying Widths (emphasis Primes)
A, or a, is the first Letter (alphabet), letter and the first vowel letter of the Latin alphabet, used in the modern English alphabet, and others worldwide. Its name in English is ''English alphabet#Letter names, a'' (pronounced ), plural ''aes''. It is similar in shape to the Ancient Greek letter alpha, from which it derives. The uppercase version consists of the two slanting sides of a triangle, crossed in the middle by a horizontal bar. The lowercase version is often written in one of two forms: the double-storey and single-storey . The latter is commonly used in handwriting and fonts based on it, especially fonts intended to be read by children, and is also found in italic type. In English, ''English articles, a'' is the indefinite article, with the alternative form ''an''. Name In English, the name of the letter is the ''long A'' sound, pronounced . Its name in most other languages matches the letter's pronunciation in open syllables. History The earliest know ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Public-key Cryptography
Public-key cryptography, or asymmetric cryptography, is the field of cryptographic systems that use pairs of related keys. Each key pair consists of a public key and a corresponding private key. Key pairs are generated with cryptographic algorithms based on mathematical problems termed one-way functions. Security of public-key cryptography depends on keeping the private key secret; the public key can be openly distributed without compromising security. There are many kinds of public-key cryptosystems, with different security goals, including digital signature, Diffie–Hellman key exchange, Key encapsulation mechanism, public-key key encapsulation, and public-key encryption. Public key algorithms are fundamental security primitives in modern cryptosystems, including applications and protocols that offer assurance of the confidentiality and authenticity of electronic communications and data storage. They underpin numerous Internet standards, such as Transport Layer Security, T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elementary Proof
In mathematics, an elementary proof is a mathematical proof that only uses basic techniques. More specifically, the term is used in number theory to refer to proofs that make no use of complex analysis. Historically, it was once thought that certain theorems, like the prime number theorem, could only be proved by invoking "higher" mathematical theorems or techniques. However, as time progresses, many of these results have also been subsequently reproven using only elementary techniques. While there is generally no consensus as to what counts as elementary, the term is nevertheless a common part of the mathematical jargon. An elementary proof is not necessarily simple, in the sense of being easy to understand or trivial. In fact, some elementary proofs can be quite complicated — and this is especially true when a statement of notable importance is involved.. Prime number theorem The distinction between elementary and non-elementary proofs has been considered especially important ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
An Introduction To The Theory Of Numbers
''An Introduction to the Theory of Numbers'' is a classic textbook in the field of number theory, by G. H. Hardy and E. M. Wright. It is on the list of 173 books essential for undergraduate math libraries. The book grew out of a series of lectures by Hardy and Wright and was first published in 1938. The third edition added an elementary proof of the prime number theorem, and the sixth edition added a chapter on elliptic curves. See also * List of important publications in mathematics This is a list of publications in mathematics, organized by field. Some reasons a particular publication might be regarded as important: *Topic creator – A publication that created a new topic *Breakthrough – A publication that cha ... References * * * * * * Reviews *E. T. Bell (July 1939). "Review: G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers", '' Bull. Amer. Math. Soc.'' 45(7): pp. 507–509 * * Citations Mathematics textbooks Number theo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harold Davenport
Harold Davenport FRS (30 October 1907 – 9 June 1969) was an English mathematician, known for his extensive work in number theory. Early life and education Born on 30 October 1907 in Huncoat, Lancashire, Davenport was educated at Accrington Grammar School, the University of Manchester (graduating in 1927), and Trinity College, Cambridge. He became a research student of John Edensor Littlewood, working on the question of the distribution of quadratic residues. First steps in research The attack on the distribution question leads quickly to problems that are now seen to be special cases of those on local zeta-functions, for the particular case of some special hyperelliptic curves such as Y^2 = X(X-1)(X-2)\ldots (X-k). Bounds for the zeroes of the local zeta-function immediately imply bounds for sums \sum \chi(X(X-1)(X-2)\ldots (X-k)), where χ is the Legendre symbol ''modulo'' a prime number ''p'', and the sum is taken over a complete set of residues mod ''p''. In the l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and in many other branches of mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers, sometimes called "the reals", is traditionally denoted by a bold , often using blackboard bold, . The adjective ''real'', used in the 17th century by René Descartes, distinguishes real numbers from imaginary numbers such as the square roots of . The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real numbers are called irrational numbers. Some irrational numbers (as well as all the rationals) a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Integer
In algebraic number theory, an algebraic integer is a complex number that is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficients are integers. The set of all algebraic integers is closed under addition, subtraction and multiplication and therefore is a commutative subring of the complex numbers. The ring of integers of a number field , denoted by , is the intersection of and : it can also be characterized as the maximal order of the field . Each algebraic integer belongs to the ring of integers of some number field. A number is an algebraic integer if and only if the ring \mathbbalpha/math> is finitely generated as an abelian group, which is to say, as a \mathbb- module. Definitions The following are equivalent definitions of an algebraic integer. Let be a number field (i.e., a finite extension of \mathbb, the field of rational numbers), in other words ... [...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 (for example, The set of all rational numbers is often referred to as "the rationals", and is closed under addition, subtraction, multiplication, and division by a nonzero rational number. It is a field under these operations and therefore also called the field of rationals or the field of rational numbers. It is usually denoted by boldface , or blackboard bold 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 n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Object
A mathematical object is an abstract concept arising in mathematics. Typically, a mathematical object can be a value that can be assigned to a Glossary of mathematical symbols, symbol, and therefore can be involved in formulas. Commonly encountered mathematical objects include numbers, Expression (mathematics), expressions, shapes, function (mathematics), functions, and set (mathematics), sets. Mathematical objects can be very complex; for example, theorems, proof (mathematics), proofs, and even theory (mathematical logic), formal theories are considered as mathematical objects in proof theory. In Philosophy of mathematics, the concept of "mathematical objects" touches on topics of existence, Identity (philosophy), identity, and the Nature (philosophy), nature of reality. In metaphysics, objects are often considered Entity, entities that possess Property (philosophy), properties and can stand in various Relation (philosophy), relations to one another. Philosophers debate whether m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 (2 × 2) 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 primality test, method of checking the primality of a given number , called trial division, tests whether is a multiple of any integer between 2 and . Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive integers Some authors acknowledge both definitions whenever convenient. Sometimes, the whole numbers are the natural numbers as well as zero. In other cases, the ''whole numbers'' refer to all of the integers, including negative integers. The counting numbers are another term for the natural numbers, particularly in primary education, and are ambiguous as well although typically start at 1. The natural numbers are used for counting things, like "there are ''six'' coins on the table", in which case they are called ''cardinal numbers''. They are also used to put things in order, like "this is the ''third'' largest city in the country", which are called ''ordinal numbers''. Natural numbers are also used as labels, like Number (sports), jersey ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
