There are several definitions of the concept of number theory. One of them says that this is a special section of mathematics (or higher arithmetic), which studies in detail integers and objects similar to them.
Another definition clarifies that this section of mathematics studies the properties of numbers and their behavior in various situations.
Some scientists believe that the theory is so vast that it is impossible to give an exact definition, but it is enough to divide it into several less voluminous theories.
It is not possible to establish reliably when the theory of numbers was born. However, it was precisely established: today the oldest, but not the only document testifying to the interest of the ancients in number theory is a small fragment of a clay tablet from 1800 BC. It contains a whole series of the so-called Pythagorean triples (natural numbers), many of which consist of five characters. A huge number of such triples excludes their mechanical selection. This indicates that interest in number theory arose, apparently, much earlier than scientists had originally assumed.
The most notable people in the development of the theory are the Pythagoreans Euclid and Diophantus, the Indians of Ariabhat, Brahmagupta and Bhaskara who lived in the Middle Ages, and even later - Fermat, Euler, Lagrange.
At the beginning of the 20th century, number theory attracted the attention of mathematical geniuses such as A. N. Korkin, E. I. Zolotarev, A. A. Markov, B. N. Delone, D. K. Faddeev, I. M. Vinogradov, G. . Weil, A. Selberg.
Developing and deepening the calculations and research of ancient mathematicians, they brought the theory to a new, much higher level, covering many areas. In-depth research and the search for new evidence led to the discovery of new problems, some of which have not been studied so far. Remain open: Artin’s hypothesis about the infinity of the set of primes, the question of the infinity of the number of primes, many other theories.
Today, the main components into which number theory is divided are theories: elementary, large numbers, random numbers, analytic, algebraic.
Elementary number theory deals with the study of integers, without involving methods and concepts from other branches of mathematics. Fibonacci numbers, Fermat’s small theorem, are the most widespread concepts even known to schoolchildren from this theory.
The theory of large numbers (or the Law of large numbers) is a subsection of probability theory that seeks to prove that the arithmetic average (in other words, the empirical average) of a large sample approaches the mathematical expectation (which is also called the theoretical average) of this sample under the condition of a fixed distribution.
The theory of random numbers, dividing all events into indefinite, deterministic and random, is trying to determine the probability of complex events by the probability of simple events. This section includes the properties of conditional probabilities and the theorem of their multiplication, the hypothesis theorem (which is often called the Bayes formula), etc.
Analytical number theory, as its name implies, uses methods and techniques of mathematical analysis to study mathematical quantities and numerical properties . One of the main directions of this theory is the proof of the theorem (using complex analysis) on the distribution of primes.
Algebraic number theory works directly with numbers, their analogues (for example, algebraic numbers), studies the theory of divisors, cohomology of groups, Dirichlet functions, etc.
Centuries-old attempts to prove Fermat's theorem led to the appearance and development of this theory.
Until the twentieth century, number theory was considered abstract science, "pure art from mathematics," having absolutely no practical or utilitarian application. Today its calculations are used in cryptographic protocols, when calculating the trajectories of satellites and space probes, in programming. Economics, finance, computer science, geology - all these sciences today are impossible without number theory.