Number Theory Concepts

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Fundamental Theorem of Arithmetic

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States that every integer greater than 1 is either a prime number itself or can be represented as a unique product of prime numbers, up to the order of the factors.

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Modular arithmetic

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A system of arithmetic for integers that works with remainders. Numbers 'wrap around' when they reach a certain value, called the modulus.

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Congruence

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Two integers a and b are congruent modulo n (written as a ≡ b (mod n)) if they have the same remainder when divided by n.

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Diophantine equation

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A polynomial equation with integer coefficients for which only integer solutions are sought.

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Multiplicative function

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An arithmetic function f(n) where f(1)=1 and f(mn)=f(m)f(n) for any two relatively prime integers m and n.

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Completely multiplicative function

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A function where f(mn)=f(m)f(n) holds for all positive integers m and n, not just coprime ones.

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Euler's totient function (φ(n))

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A multiplicative function that counts the number of positive integers up to a given integer n that are relatively prime to n.

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Möbius function (μ(n))

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A multiplicative function used in number theory, particularly in the Möbius inversion formula, which relates two multiplicative functions.

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Algebraic number theory

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The study of algebraic integers, which are the roots of monic polynomials with rational integer coefficients.

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Analytic number theory

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A branch of number theory that uses methods from mathematical analysis to solve problems about integers, such as those concerning the distribution of prime numbers.