Számelmélettel

Számelmélettel, or number theory, is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. It is concerned with the properties of the integers, such as divisibility, prime numbers, congruences, and Diophantine equations.

Historically, number theory has been divided into several subfields. Elementary number theory is concerned with the integers using elementary techniques. Analytic number theory uses tools from mathematical analysis to study problems about integers. Algebraic number theory studies the properties of integers by passing to algebraic structures, such as number fields. Probabilistic number theory studies the distribution of number-theoretic objects using probabilistic methods.

Key concepts in number theory include prime numbers, which are integers greater than 1 that have no positive divisors other than 1 and themselves. The distribution of prime numbers is a central theme, investigated through the Prime Number Theorem. Congruence arithmetic, introduced by Carl Friedrich Gauss, deals with remainders upon division and is fundamental to many areas of number theory and cryptography.

Another important area is Diophantine equations, which are polynomial equations for which integer solutions are sought. For example, Fermat's Last Theorem, which states that no three positive integers a, b, and c can satisfy the equation a^n + b^n = c^n for any integer value of n greater than 2, was a famous problem in this area.

Number theory has applications in various fields, including cryptography, computer science, and coding theory. Its abstract nature often leads to unexpected connections with other areas of mathematics and science.