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A Lehmer sieve, which is a primitive digital computer once used for finding primes and solving simple Diophantine equations.

Number theory or arithmetic<ref group=note>Especially in older sources; see two following notes.</ref> is a branch of pure mathematics devoted primarily to the study of the natural numbers and the integers. It is sometimes called "The Queen of Mathematics" because of its foundational place in the discipline.{{#invoke:Footnotes|sfn}} Number theorists study prime numbers as well as the properties of objects made out of integers (e.g., rational numbers) or defined as generalizations of the integers (e.g., 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 (e.g., 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, e.g., 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 By the early twentieth century, it had been superseded by "number theory".<ref group=note>Already in 1921, T. L. Heath had to explain: "By arithmetic, Plato meant, not arithmetic in our sense, but the science which considers numbers in themselves, in other words, what we mean by the Theory of Numbers." </ref> (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.) 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.<ref group=note>Take, e.g. {{#invoke:Footnotes|harvard_citation_no_bracket}}. In 1952, Davenport still had to specify that he meant The Higher Arithmetic. Hardy and Wright wrote in the introduction to An Introduction to the Theory of Numbers (1938): "We proposed at one time to change [the title] to An introduction to arithmetic, a more novel and in some ways a more appropriate title; but it was pointed out that this might lead to misunderstandings about the content of the book." </ref> In particular, arithmetical is preferred as an adjective to number-theoretic.


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