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Introduction to Number Theory

The Number theory is a group of pure mathematics devoted mainly to the study of the integers. The number theorists study prime numbers with properties of objects made out of integers (example, rational numbers) or defined as generalizations of the integers (example, algebraic integers).

The integers can be considered either in themselves or as answers to equations (i.e., diophantine geometry). The Questions of number theory are frequently best understood through the study of analytical objects (example, the Riemann zeta functions) which encode properties of integers, primes, or the other number-theoretic objects in some fashion. One can also study the real numbers in relation to rational numbers, illustration, as approximated by the latter diophantine approximation.

The historical name of number theory is arithmetic. By the early 20^{th} century, it had been superseded by "number theory"(The term "arithmetic" is used usually to mean "elementary calculations"; it has also obtained other meanings in mathematical logic similar to as in Peano arithmetic, and computer science similar, as in floating point arithmetic). The employ of the term arithmetic for number theory recovered some ground in the second half of the 20th century, arguably in part as of French influence. Mainly, arithmetical is termed as an adjective to number-theoretic.

**Main subdivisions:**

**Analytic number theory:**

The analytic number theory might be defined in terms of its tools, similar to the study of the integers by means of tools from real and complex analysis; or in terms of its concerns, as the study in number theory of estimations on density and size, as opposed to identities. A few subjects commonly considered to be section of analytic number theory, illustration, sieve theory, are better covered by the second instead of the first definition: a few of sieve theory, for illustration, uses little analysis yet it is considered to be section of analytic number theory.

The following are illustrations of problems in analytic number theory: the prime number theorem, the Goldbach conjectures the waring problem and the Riemann Hypothesis. A few of the most necessary tools of analytic number theory are the circle technique, sieve techniques and L-functions (or, instead, the study of their properties). The theories of modular forms (and, more generally, automorphic forms) also occupy an increasingly central place in the toolbox of the analytic number theory.

**Algebraic number theory:**

The algebraic number theory studies algebraic objects of interest and algebraic properties in number theory. (Therefore, analytic and algebraic number theory can and do overlap: the previous is defined by its techniques, the later by its objects of study.) The main topic is that of the algebraic numbers that generalizations of the rational number are. In short, an algebraic number is any complex number which is a solution to some polynomial equation with rational coefficients; for example, every solution is an algebraic number. The fields of algebraic numbers are also termed as shortly number fields, or the algebraic number fields.

It could be argued that the very easy kind of number fields (viz., quadratic fields) were formerly studied by Gauss, as the discussion of quadratic forms in Disquisitiones arithmetical can be restated in terms of standards and norms in quadratic fields. For that matter, 11^{th} century chakravala technique amounts-in modern terms-to an algorithm for finding the units of a real quadratic number field. So far, neither Bhaskara nor Gauss knew of number fields as such.

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