Foundations and History of Number Theory

Definition and Scope of Number Theory

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.

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Integers and Their Properties

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