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t. e. 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. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." [1]
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields.
A number x is called an A*-number if the ω*(x, n) converge to 0. If the ω*( x , n ) are all finite but unbounded, x is called a T*-number , Koksma's and Mahler's classifications are equivalent in that they divide the transcendental numbers into the same classes. [ 32 ]
1. The class number of a number field is the cardinality of the ideal class group of the field. 2. In group theory, the class number is the number of conjugacy classes of a group. 3. Class number is the number of equivalence classes of binary quadratic forms of a given discriminant. 4.
Analytic number theory. Riemann zeta function ζ (s) in the complex plane. The color of a point s encodes the value of ζ (s): colors close to black denote values close to zero, while hue encodes the value's argument. In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve ...
In mathematics and computer science, computational number theory, also known as algorithmic number theory, is the study of computational methods for investigating and solving problems in number theory and arithmetic geometry, including algorithms for primality testing and integer factorization, finding solutions to diophantine equations, and explicit methods in arithmetic geometry. [1]
Geometry of numbers is the part of number theory which uses geometry for the study of algebraic numbers. Typically, a ring of algebraic integers is viewed as a lattice in R n , {\displaystyle \mathbb {R} ^{n},} and the study of these lattices provides fundamental information on algebraic numbers. [ 1 ]
1937 — I. M. Vinogradov proves Vinogradov's theorem that every sufficiently large odd integer is the sum of three primes, a close approach to proving Goldbach's weak conjecture. 1949 — Atle Selberg and Paul Erdős give the first elementary proof of the prime number theorem. 1966 — Chen Jingrun proves Chen's theorem, a close approach to ...