Number theory (or arithmetic, in older terms higher arithmetic) is a branch of pure mathematics devoted primarily to the study of integers and integer functions. Mathematics is the queen of sciences, and number theory is the queen of mathematics. Number theorists refer to prime numbers and mathematical objects (such as algebraic integers) that either consist of integers (such as rational numbers) or are defined as generalizations of integers. We study the properties of Integers can be viewed either in themselves or as solutions to equations (Diophantine geometry).Arithmetic problems often involve integers, prime numbers, or other arithmetic It is best understood by studying analytic objects (e.g. Riemann's zeta function) that somehow encode the properties of objects (analytic number theory). You can also study real numbers in relation to rational numbers. For example, how it is approximated by rational numbers (Diophantine approximation). The old term for number theory is arithmetic. At the beginning of the 20th century it was superseded by "number theory". (The word "arithmetic" is used by the general public for "basic computation". It has also acquired other meanings in mathematical logic, such as Peano arithmetic, and in computer science, such as floating-point arithmetic. ) The term arithmetic in number theory regained momentum in the second half of the 20th century, probably due in part to French influence. In particular, arithmetic is generally the preferred adjective for number theory. Traditionally, number theory is the branch of mathematics concerned with the properties of integers, and many of its unsolved problems are easily understood by non-mathematicians. More generally, this field deals with a broader class of problems that naturally arise from studying integers. Number theory can be divided into several areas, depending on the methods used and the problems examined although the closed-form formula for the splitting function is not known; there are both an asymptotic expansion that approximates it exactly, and an iteration relation that can be computed exactly. It grows as an exponential function of the square root of the argument. The multiplicative inverse of that generating function is the Euler function. By Euler's pentagon theorem, this function is the alternating sum of the pentagon powers of its arguments. Algebraic number theory is a branch of number theory that uses abstract algebraic techniques to study integers, rational numbers, and their generalizations. Arithmetic problems are expressed in terms of properties of algebraic objects such as algebraic number fields and their integer rings, finite fields, and function fields. These properties, such as whether the ring admits unique factorization, ideal behaviour, and the Galois group of the field, can solve fundamentally important problems in number theory, such as the existence of solutions to the Diophantine equation. The theory of functions of one complex variable, the historical name of complex analysis, the branch of mathematical analysis that studies functions of complex numbers. Constitutive Function Theory, the study of the relationship between the smoothness of a function and its degree of approximation. The best-known application of number theory is public-key cryptography, which is: B. RSA Algorithm. Second, public key cryptography enables many technologies that we take for granted. B. Ability to conduct secure online transactions.

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