The Number Theory
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The Number Theory
by showmyiq » Thu Oct 18, 2012 11:43 pm
The classical number theory is the branch of mathematics that examines the properties of integers. From the relatively more number theory deals with a wider class of problems that naturally occur in the study of integers. It is divided into several sub regions, depending on the methods used and the types of issues being considered.
The term "arithmetic" is sometimes used as a synonym of "number theory". This is a relatively old matter and is not very popular. Number theory used to be called higher arithmetic, but this term also has dropped out of use. However, still participate in some names (arithmetic functions, arithmetic of elliptic curves, fundamental theorem of arithmetic). This meaning of the term arithmetic should not be confused with elementary arithmetic.
Elementary number theory
In elementary number theory, integers are studied without using methods from other areas of mathematics. It deals with issues such as divisibility, use of the Euclidean algorithm for finding the greatest common divisor, decomposition of integers as product of simple, perfect numbers, comparisons and more. Some of the important discoveries in this area include: little theorem of Fermat, Euler's theorem, the theorem and the law of quadratic reciprocity. Studying the properties of multiplicative functions such as Mobius function and Euler's function, integer sequences, factorials and Fibonacci numbers also fall in this area.
A lot of theoretic problems can speak in terms of basic theory but nonetheless require much deeper research and methods from other areas of mathematics. For Example:
• The hypothesis of the Conjecture which asserts that every even number greater than two can be expressed as the sum of two simple.
• The hypothesis of prime numbers twins, which claims that there are countless pairs of primes with difference 2, 3 and 5 or 11 and 13.
• The big theorem of Farm (which is spoken in 1637, but was proven right in 1994) which claims that the equation x^n + y^n = z^n has a solution in integers for n greater than 2
Analytic number theory
Analytic number theory uses the resources of the differential and integral calculus and complex analysis, to answer questions about integers. The law of distribution of prime numbers and the related Riemann hypothesis are areas of analytic number theory. Other problems include: the problem of Waring (concerning the presentation of a given integer as a sum of squares, cubes, etc.), the hypothesis of twin prime numbers and hypothesis of Conjecture. Proof that π and e are transcendental, also belong to the analytic number theory. Although allegations of transcendental numbers at first glance do not have much to do with the study of integers, they actually study the possible values of polynomials with coefficients calculated for example in number e.They are also related to the theory of Diophantine approximations, which examines how well a given real number may be closer to a rational number.
Algebraic number theory
In algebraic number theory, the concept of number has been extended to the algebraic numbers that are roots of polynomials with rational coefficients. They contain elements analogous to the integers, the so-called purpose algebraic numbers. The known properties of the target (for example, digital scanning of simple multipliers) is not always preserved.
History
Ancient Greece
Number theory is a popular topic for ancient Greek mathematicians of the late Hellenistic era. In the 3rd century in Alexandria they already examine some special cases of Diophantine equations, which later received its name from Diophantus. He makes and attempts to find integer solutions to linear indeterminate equations, such as x+y = 5. Diophantus discovered that many unspecified equations can be reduced to a form for that particular category of solutions is known, although a specific decision is not.
India
Diophantine equations are studied in India in the middle of the first Millennium. There were created the first systematic methods for setting goals roots of Diophantine equations. In 499 year in its composition "Ariabhatia", Aryabhatta makes the first explicit description of the General integer solution of linear Diophantine’s equation ay + bx = c. His method, considered his most significant contributions to pure mathematics, based on the use of chainsaws and fractions. Aryabhatta used to solve systems of linear Diophantine equations, used in astronomy.
In the year 628 Brahmagupta uses the method “chakraval” to solve Diophantine square equations, including variants of the Pell equation as 61x^2 + 1 = y^2. His works have been translated into Arabic in 773 and year of Latin in 1126 year through its influence on Islamic and Western theory. In the year 1150 Bhaskara II apply a modified form of the method “chakraval” and reaches a total solution of the Pell equation. A similar solution was found again in Europe until the 18th century.
Early modern history
Number theory is reborn in the sixteenth and seventeenth centuries in Europe, with François Viète, Bachet de Meziriac, and especially Fermat, whose method of infinite descent is a first general idea for solving Diophantine equations. Major contributions in the eighteenth century made Euler and Lagrange.
Home of systematic theory
Around the beginning of the nineteenth century books of Legendre (1798), and Gauss put the first systematic theories. It can be said that the book Disquisitiones Arithmeticae Gauss (1801) marks the beginning of the modern theory of numbers.
Gauss first formulated the theory of comparisons. It introduces the name -> mod (variale) and explores the vast majority of this area. Čebišev in 1847 published a work in Russian.
Besides summarizing previous results, Legendre published the law of quadratic reciprocity. This law, originally formulated by Euler is first attested in the book of Legendre Théorie des Nombres (1798) for some private cases. Regardless of them, Gauss rediscovered the law in 1795, and was the first that gives a general proof. To the subject also contribute: Cauchy, Dirichlet, Jacobi, who introduced the Jacobi symbol; Liouville, Eisenstein, Kummer, and Kronecker. The theory is extended to include bi and cubic reciprocity.
Theory of prime numbers
Prolific theme in number theory is the study of the distribution of prime numbers. Even as a teenager, Gauss formulates the hypothesis concerning the number of prime numbers, which do not exceed a number (see the law of distribution of prime numbers).
Čebišev (1850) gives useful constraints on the top and bottom of the same number. Riemann first used complex analysis in the study of the Zeta function. This leads to the finding of a link between the zeros of the Zeta function and the distribution of prime numbers, which in the end gives a proof of the law of distribution of prime numbers, found by Adamand and Valle-Poussin in 1896. Later, in 1969, Paul Erd′oš and Atle Selberg gave an elementary proof of the theorem. Here elementary means that it does not use techniques from complex analysis proof, however, is quite difficult. The Riemann hypothesis, would give much more accurate information on the issue, but it is not yet known whether it is correct.
The term "arithmetic" is sometimes used as a synonym of "number theory". This is a relatively old matter and is not very popular. Number theory used to be called higher arithmetic, but this term also has dropped out of use. However, still participate in some names (arithmetic functions, arithmetic of elliptic curves, fundamental theorem of arithmetic). This meaning of the term arithmetic should not be confused with elementary arithmetic.
Elementary number theory
In elementary number theory, integers are studied without using methods from other areas of mathematics. It deals with issues such as divisibility, use of the Euclidean algorithm for finding the greatest common divisor, decomposition of integers as product of simple, perfect numbers, comparisons and more. Some of the important discoveries in this area include: little theorem of Fermat, Euler's theorem, the theorem and the law of quadratic reciprocity. Studying the properties of multiplicative functions such as Mobius function and Euler's function, integer sequences, factorials and Fibonacci numbers also fall in this area.
A lot of theoretic problems can speak in terms of basic theory but nonetheless require much deeper research and methods from other areas of mathematics. For Example:
• The hypothesis of the Conjecture which asserts that every even number greater than two can be expressed as the sum of two simple.
• The hypothesis of prime numbers twins, which claims that there are countless pairs of primes with difference 2, 3 and 5 or 11 and 13.
• The big theorem of Farm (which is spoken in 1637, but was proven right in 1994) which claims that the equation x^n + y^n = z^n has a solution in integers for n greater than 2
Analytic number theory
Analytic number theory uses the resources of the differential and integral calculus and complex analysis, to answer questions about integers. The law of distribution of prime numbers and the related Riemann hypothesis are areas of analytic number theory. Other problems include: the problem of Waring (concerning the presentation of a given integer as a sum of squares, cubes, etc.), the hypothesis of twin prime numbers and hypothesis of Conjecture. Proof that π and e are transcendental, also belong to the analytic number theory. Although allegations of transcendental numbers at first glance do not have much to do with the study of integers, they actually study the possible values of polynomials with coefficients calculated for example in number e.They are also related to the theory of Diophantine approximations, which examines how well a given real number may be closer to a rational number.
Algebraic number theory
In algebraic number theory, the concept of number has been extended to the algebraic numbers that are roots of polynomials with rational coefficients. They contain elements analogous to the integers, the so-called purpose algebraic numbers. The known properties of the target (for example, digital scanning of simple multipliers) is not always preserved.
History
Ancient Greece
Number theory is a popular topic for ancient Greek mathematicians of the late Hellenistic era. In the 3rd century in Alexandria they already examine some special cases of Diophantine equations, which later received its name from Diophantus. He makes and attempts to find integer solutions to linear indeterminate equations, such as x+y = 5. Diophantus discovered that many unspecified equations can be reduced to a form for that particular category of solutions is known, although a specific decision is not.
India
Diophantine equations are studied in India in the middle of the first Millennium. There were created the first systematic methods for setting goals roots of Diophantine equations. In 499 year in its composition "Ariabhatia", Aryabhatta makes the first explicit description of the General integer solution of linear Diophantine’s equation ay + bx = c. His method, considered his most significant contributions to pure mathematics, based on the use of chainsaws and fractions. Aryabhatta used to solve systems of linear Diophantine equations, used in astronomy.
In the year 628 Brahmagupta uses the method “chakraval” to solve Diophantine square equations, including variants of the Pell equation as 61x^2 + 1 = y^2. His works have been translated into Arabic in 773 and year of Latin in 1126 year through its influence on Islamic and Western theory. In the year 1150 Bhaskara II apply a modified form of the method “chakraval” and reaches a total solution of the Pell equation. A similar solution was found again in Europe until the 18th century.
Early modern history
Number theory is reborn in the sixteenth and seventeenth centuries in Europe, with François Viète, Bachet de Meziriac, and especially Fermat, whose method of infinite descent is a first general idea for solving Diophantine equations. Major contributions in the eighteenth century made Euler and Lagrange.
Home of systematic theory
Around the beginning of the nineteenth century books of Legendre (1798), and Gauss put the first systematic theories. It can be said that the book Disquisitiones Arithmeticae Gauss (1801) marks the beginning of the modern theory of numbers.
Gauss first formulated the theory of comparisons. It introduces the name -> mod (variale) and explores the vast majority of this area. Čebišev in 1847 published a work in Russian.
Besides summarizing previous results, Legendre published the law of quadratic reciprocity. This law, originally formulated by Euler is first attested in the book of Legendre Théorie des Nombres (1798) for some private cases. Regardless of them, Gauss rediscovered the law in 1795, and was the first that gives a general proof. To the subject also contribute: Cauchy, Dirichlet, Jacobi, who introduced the Jacobi symbol; Liouville, Eisenstein, Kummer, and Kronecker. The theory is extended to include bi and cubic reciprocity.
Theory of prime numbers
Prolific theme in number theory is the study of the distribution of prime numbers. Even as a teenager, Gauss formulates the hypothesis concerning the number of prime numbers, which do not exceed a number (see the law of distribution of prime numbers).
Čebišev (1850) gives useful constraints on the top and bottom of the same number. Riemann first used complex analysis in the study of the Zeta function. This leads to the finding of a link between the zeros of the Zeta function and the distribution of prime numbers, which in the end gives a proof of the law of distribution of prime numbers, found by Adamand and Valle-Poussin in 1896. Later, in 1969, Paul Erd′oš and Atle Selberg gave an elementary proof of the theorem. Here elementary means that it does not use techniques from complex analysis proof, however, is quite difficult. The Riemann hypothesis, would give much more accurate information on the issue, but it is not yet known whether it is correct.
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