information:filtering, stylistic:tone, structural:reformatting
Number theory

Number theory studies the structure and properties of integers as well as the relations and laws between them. Some of the main branches of modern number theory include elementary number theory, analytic number theory, algebraic number theory, and geometric number theory. Elementary number theory studies aspects of integers that can be investigated using elementary methods. Its topics include divisibility, factorization, and primality. Analytic number theory, by contrast, relies on techniques from analysis and calculus. It examines problems like how prime numbers are distributed and the claim that every even number is a sum of two prime numbers. Algebraic number theory employs algebraic structures to analyze the properties of and relations between numbers. Examples are the use of fields and rings, as in algebraic number fields like the ring of integers. Geometric number theory uses concepts from geometry to study numbers. For instance, it investigates how lattice points with integer coordinates behave in a plane. Further branches of number theory are probabilistic number theory, which employs methods from probability theory, combinatorial number theory, which relies on the field of combinatorics, computational number theory, which approaches number-theoretic problems with computational methods, and applied number theory, which examines the application of number theory to fields like physics, biology, and cryptography.

Influential theorems in number theory include the fundamental theorem of arithmetic, Euclid's theorem, and Fermat's last theorem. According to the fundamental theorem of arithmetic, every integer greater than 1 is either a prime number or can be represented as a unique product of prime numbers. For example, the number 18 is not a prime number and can be represented as 2 × 3 × 3 {\displaystyle 2\times 3\times 3}, all of which are prime numbers. The number 19, by contrast, is a prime number that has no other prime factorization. Euclid's theorem states that there are infinitely many prime numbers. Fermat's last theorem is the statement that no positive integer values can be found for a {\displaystyle a}, b {\displaystyle b}, and c {\displaystyle c}, to solve the equation a n + b n = c n {\displaystyle a^{n}+b^{n}=c^{n}} if n {\displaystyle n} is greater than 2 {\displaystyle 2}.