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Number theory primarily studies integers, their properties, and the relationships among them.
Number theory is a branch of pure mathematics focused primarily on the integers and their properties, relationships, and arithmetic functions. It studies prime numbers, divisibility, factorization, congruences, integer-valued functions, and equations whose solutions are integers or rational numbers. The field also examines mathematical structures constructed from or generalizing the integers, such as rational numbers and algebraic integers. The scope of number theory includes several specialized areas. Elementary number theory uses basic arithmetic methods to study topics such as primes, greatest common divisors, Diophantine equations, and modular arithmetic. Analytic number theory applies calculus, complex analysis, and functions such as the Riemann zeta function; algebraic number theory uses structures such as rings and fields; and geometric, probabilistic, combinatorial, computational, and applied number theory use other methods or address practical applications. Although traditionally regarded as pure mathematics, number theory has important modern applications, especially in cryptography and computer science.
Number theory primarily studies integers, their properties, and the relationships among them.
Important subjects include prime numbers, divisibility, factorization, congruences, arithmetic functions, and Diophantine equations.
The field extends beyond integers to related or generalized objects such as rational numbers and algebraic integers.
Major subdivisions include elementary, analytic, algebraic, geometric, probabilistic, combinatorial, computational, and applied number theory.
Number theory has both theoretical significance and practical applications, including public-key cryptography.
A whole number that includes positive numbers, negative numbers, and zero.
An integer greater than one whose only positive divisors are one and itself.
A relationship in which one integer divides another without leaving a remainder.
A function defined on the positive integers that assigns values according to number-theoretic properties.
A polynomial equation for which integer or rational solutions are sought.
A system of arithmetic that studies remainders and congruences relative to a fixed positive integer called the modulus.
The study of number-theoretic problems using basic arithmetic methods.
The study of number theory using tools from calculus, complex analysis, and related analytical methods.
The study of numbers and their relationships using algebraic structures such as rings and fields.
The use of number-theoretic concepts in areas such as cryptography, computer science, and technology.
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