Number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. 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 can often be understood through the study of analytical objects, such as 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, as for instance how irrational numbers can be approximated by fractions (Diophantine approximation).

Number theory is one of the oldest branches of mathematics alongside geometry. One quirk of number theory is that it deals with statements that are simple to understand but are very difficult to solve. Examples of this are Fermat's Last Theorem, which was proved 358 years after the original formulation, and Goldbach's conjecture, which remains unsolved since the 18th century. German mathematician Carl Friedrich Gauss (1777–1855) once remarked, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." It was regarded as the example of pure mathematics with no applications outside mathematics until the 1970s, when it became known that prime numbers would be used as the basis for the creation of public-key cryptography algorithms.

Number theory is the branch of mathematics that studies integers and their properties and relations. The integers comprise a set that extends the set of natural numbers ${\displaystyle \{1,2,3,\dots \}}$ to include number ${\displaystyle 0}$ and the negation of natural numbers ${\displaystyle \{-1,-2,-3,\dots \}}$. 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).

Number theory is closely related to arithmetic and some authors use the terms as synonyms. However, the word "arithmetic" is used today to mean the study of numerical operations and extends to the real numbers. In a more specific sense, number theory is restricted to the study of integers and focuses on their properties and relationships. Traditionally, it is known as higher arithmetic. By the early twentieth century, the term number theory had been widely adopted. The term number means whole numbers, which refers to either the natural numbers or the integers.

Elementary number theory studies aspects of integers that can be investigated using elementary methods such as elementary proofs. Analytic number theory, by contrast, relies on complex numbers and techniques from analysis and calculus. Algebraic number theory employs algebraic structures such as fields and rings to analyze the properties of and relations between numbers. Geometric number theory uses concepts from geometry to study numbers. Further branches of number theory are probabilistic number theory, combinatorial number theory, computational number theory, and applied number theory, which examines the application of number theory to science and technology.

In recorded history, knowledge of numbers existed in the ancient civilisations of Mesopotamia, Egypt, China, and India. The earliest historical find of an arithmetical nature is the Plimpton 322, dated c. 1800 BC. It is a broken clay tablet that contains a list of Pythagorean triples, that is, integers ${\displaystyle (a,b,c)}$ such that ${\displaystyle a^{2}+b^{2}=c^{2}}$. The triples are too numerous and too large to have been obtained by brute force. The table's layout suggests that it was constructed by means of what amounts, in modern language, to the identity$${\displaystyle \left({\frac {1}{2}}\left(x-{\frac {1}{x}}\right)\right)^{2}+1=\left({\frac {1}{2}}\left(x+{\frac {1}{x}}\right)\right)^{2},}$$which is implicit in routine Old Babylonian exercises. It has been suggested instead that the table was a source of numerical examples for school problems. Plimpton 322 tablet is the only surviving evidence of what today would be called number theory within Babylonian mathematics, though a kind of Babylonian algebra was much more developed.

Although other civilizations probably influenced Greek mathematics at the beginning, all evidence of such borrowings appear relatively late, and it is likely that Greek arithmētikḗ (the theoretical or philosophical study of numbers) is an indigenous tradition. The ancient Greeks developed a keen interest in divisibility. The Pythagoreans attributed mystical quality to perfect and amicable numbers. The Pythagorean tradition also spoke of so-called polygonal or figurate numbers. Euclid devoted part of his Elements to topics that belong to elementary number theory, including prime numbers and divisibility. He gave the Euclidean algorithm for computing the greatest common divisor of two numbers and a proof implying the infinitude of primes. The foremost authority in arithmētikḗ in Late Antiquity was Diophantus of Alexandria, who probably lived in the 3rd century AD. He wrote Arithmetica, a collection of worked-out problems where the task is invariably to find rational solutions to a system of polynomial equations, usually of the form ${\displaystyle f(x,y)=z^{2}}$ or ${\displaystyle f(x,y,z)=w^{2}}$. In modern parlance, Diophantine equations are polynomial equations to which rational or integer solutions are sought.