There are several definitions of "theory of numbers."One of them says that a special branch of mathematics (arithmetic or higher), which examines in detail the integers and objects similar to them.
Another definition specifies that this branch of mathematics studying the properties of numbers and their behavior in different situations.
Some scientists believe that the theory is so vast that it give a precise definition is impossible, and you just split into several less volume theories.
Set reliably when originated the theory of numbers is not possible.However, well established: as of today the oldest, but not the only document testifying to the interest of the ancient theory of numbers, is a small fragment of a clay tablet 1800s BC.In it - a number of so-called Pythagorean triples (positive integers), many of which are made up of five characters.A huge number of such triples excludes their mechanical selection.This suggests that interest in number theory came, apparently, much earlier than scientists originally expected.
most prominent actors in the development of the theory of the Pythagoreans considered Euclid and Diophantus, who lived in the Middle Ages Indians Aryabhata, Bhaskara and Brahmagupta, and later - Fermat, Euler, Lagrange.
In the early twentieth century, number theory has attracted the attention of such mathematical geniuses like Korkin, EI Zolotarev, Markov, Delone, DK Faddeev, Vinogradov, Weyl, Selberg.
developing and deepening the calculations and studies of ancient mathematicians, they brought the theory to a new, much higher level, covering many areas.Deep research and search for new evidence and led to the discovery of new problems, some of which have not been studied until now.Remain open: Artin's conjecture on the infinite set of primes, the question of the infinite number of prime numbers, many other theories.
At present the main components, which are divided into number theory, a theory: elementary, large numbers of random numbers, analytical, algebraic.
elementary number theory deals with the study of integers, without drawing techniques and concepts from other branches of mathematics.Fibonacci numbers, Fermat's Little Theorem - that is the most common, well-known even to schoolchildren concepts from this theory.
theory of large numbers (or the law of large numbers) - subsection probability theory, seeks to prove that the arithmetic average (on another - the average of thumb) large sample of close to expectation (which is also called the theoretical average) of this sample provided a fixed distribution.
theory of random numbers, separating all the events on the vague, deterministic and random, trying to determine the probability of the likelihood of simple events difficult.This section includes the properties of conditional probability theorem of multiplication theorem hypotheses (often called Bayes' formula), and so forth.
analytic number theory, as is clear from its name, for the study of mathematical quantities and numerical properties of the methods and techniques of mathematical analysis.One of the main directions of this theory - the proof (using complex analysis) on the distribution of prime numbers.
Algebraic Number Theory works directly with the numbers of their peers (eg, algebraic numbers), studying the theory of divisors, cohomology groups, the Dirichlet function, etc.
to the emergence and development of this theory led centuries-old attempts to prove Fermat's theorem.
Until the twentieth century, number theory was considered as an abstract science, "pure art of mathematics", do not have absolutely no practical or utilitarian use.Today, it is used in the computation of cryptographic protocols in calculating the trajectories of satellites and space probes in programming.Economics, finance, computer science, geology - all these sciences today are impossible without the theory of numbers.
