"Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions."
The study of the properties and relationships of positive integers and related concepts like divisibility, prime numbers, GCD, LCM, etc.
Divisibility: Study of how numbers can be divided evenly without leaving a remainder.
Prime numbers: Numbers that can only be divided by 1 and itself without leaving a remainder.
Composite numbers: Numbers that can be divided by more than two numbers.
Greatest common divisor (GCD): The largest positive integer that divides two or more integers without a remainder.
Relative primes (coprimes): Two numbers that do not have any common divisors other than 1.
Euclidean algorithm: An algorithm for finding the GCD of two numbers.
Fundamental theorem of arithmetic: Every integer greater than 1 can be written uniquely as a product of primes.
Modular arithmetic: A system of arithmetic for integers, where numbers wrap around after reaching a certain value (the modulus).
Congruence: Two numbers are congruent if their difference is divisible by a fixed number (the modulus).
Euler's totient function: The number of positive integers less than or equal to n that are relatively prime to n.
Fermat's little theorem: A theorem that helps simplify modular exponentiation.
Chinese remainder theorem: A theorem that provides a solution to a system of linear congruences.
Quadratic residues: Numbers that are possibly a square (mod p).
Euler's criterion: A theorem that tests if a quadratic residue is a square.
Wilson's theorem: A theorem that provides a necessary and sufficient condition for a number to be prime.
Modular inverse: The inverse of a number modulo another number.
"German mathematician Carl Friedrich Gauss (1777–1855) said, 'Mathematics is the queen of the sciences—and number theory is the queen of mathematics.'"
"Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for example, algebraic integers)."
"Integers can be considered either in themselves or as solutions to equations (Diophantine geometry)."
"Questions in number theory are often best understood through the study of analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory)."
"One may also study real numbers in relation to rational numbers, for example, as approximated by the latter (Diophantine approximation)."
"The older term for number theory is arithmetic."
"By the early twentieth century, it had been superseded by 'number theory'."
"The word 'arithmetic' is used by the general public to mean 'elementary calculations'."
"The use of the term arithmetic for number theory regained some ground in the second half of the 20th century, arguably in part due to French influence."
"In particular, arithmetical is commonly preferred as an adjective to number-theoretic."
"Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for example, algebraic integers)."
"Questions in number theory are often best understood through the study of analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory)."
"One may also study real numbers in relation to rational numbers, for example, as approximated by the latter (Diophantine approximation)."
"The word 'arithmetic' is used by the general public to mean 'elementary calculations'; it has also acquired other meanings in mathematical logic, as in Peano arithmetic, and computer science, as in floating-point arithmetic."
"Questions in number theory are often best understood through the study of analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory)."
"By the early twentieth century, it had been superseded by 'number theory'."
"In particular, arithmetical is commonly preferred as an adjective to number-theoretic."
"Integers can be considered either in themselves or as solutions to equations (Diophantine geometry)."
"German mathematician Carl Friedrich Gauss (1777–1855) said, 'Mathematics is the queen of the sciences—and number theory is the queen of mathematics.'"