Number theory: theory and practice

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 the integers and objects similar to them.

Another definition specifies that this section of mathematics studies the properties of numbers and their behavior in different situations.

Some scientists believe that the theory is so vast that it is impossible to give its exact definition, but it is enough to divide it into somewhat less voluminous theories.

It is not possible to establish reliably when the theory of numbers was born. However, it is precisely established: for today the oldest, but not the only document, indicative of the interest of the ancients in the theory of numbers, is a small fragment of the clay tablet of the 1800s before our era. In it - a number of so-called Pythagorean triples (natural numbers), many of which consist of five signs. A huge number of such triples excludes their mechanical selection. This indicates that interest in the theory of numbers arose, apparently, much earlier than originally assumed by scientists.

The most notable figures in the development of the theory are the Pythagoreans Euclid and Diophantus, the Indians of Aryabhata, Brahmagupta and Bhaskara, and later, Ferma, Euler, Lagrange.

In the early twentieth century, number theory attracted the attention of such mathematical geniuses as AN Korkin, EI Zolotarev, AA Markov, BN Delone, DK Faddeev, IM Vinogradov, G . Veil, A. Selberg.

Developing and deepening the calculations and studies of ancient mathematicians, they brought the theory to a new, much higher level, encompassing a multitude of areas. Deep research and the search for new evidence led to the discovery of new problems, some of which have not been studied so far. Open are: Artin's conjecture about the infinity of the set of primes, the question of the infinity of the number of primes, and many other theories.

To date, the main components, divided by number theory, are theories: elementary, large numbers, random numbers, analytical, algebraic.

Elementary number theory deals with the study of integers, without involving methods and concepts from other sections of mathematics. Fibonacci numbers, Fermat's small theorem, are the most common concepts, known to students, from this theory.

The theory of large numbers (or the Law of Large Numbers) is a subsection of probability theory, which seeks to prove that the average of the arithmetic mean of a large sample (in other words, the average empirical one) is close to the mathematical expectation (which is also called the theoretical mean) of this sample under the condition of a fixed distribution.

The theory of random numbers, dividing all events into indeterminate, deterministic and random, tries to determine the probability of simple events by the probability of complex ones. This section includes the properties of conditional probabilities and the theorem of their multiplication. A hypothesis theorem (which is often called the Bayes formula), etc.

Analytic number theory, as its name implies, uses methods and methods 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) about the distribution of prime numbers.

Algebraic number theory works directly with numbers, their analogs (for example, algebraic numbers), studies the theory of divisors, cohomology groups, Dirichlet functions, and so on.

The emergence and development of this theory resulted in centuries-old attempts to prove Fermat's theorem.

Until the twentieth century, the theory of numbers was considered an abstract science, "pure art from mathematics," which had absolutely no practical or utilitarian application. Today, its computations are used in cryptographic protocols, in calculating the trajectories of satellites and space probes, in programming. Economics, finance, computer science, geology - all these sciences are impossible today without the theory of numbers.