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Number theory

Number theory is the branch of [pure mathematics] concerned with the properties of [number] s in general, and [integer] s in particular, as well as the wider classes of problems that arise from their study.

Number theory may be subdivided into several fields, according to the methods used and the type of questions investigated. ( See the [list of number theory topics] .)

The terms " [arithmetic] " or "the higher arithmetic" as [nouns] are also used to refer to number theory. These are somewhat older terms, which are no longer as popular as they once were. However the word "arithmetic" is popularly used as an [adjective] rather than the more cumbersome phrase "number-theoretic", and also "arithmetic of" rather than "number theory of", e.g. [arithmetic geometry] , [arithmetic function] s, [arithmetic of elliptic curves] .

When arranging the natural numbers in a spiral and emphasizing the prime numbers, an intriguing and not fully explained pattern is observed, called the Ulam spiral.

Elementary number theory
In elementary number theory , integers are studied without use of techniques from other mathematical fields. Questions of [divisibility] , use of the [Euclidean algorithm] to compute [greatest common divisor] s, [integer factorization] s into [prime number] s, investigation of [perfect number] s and [congruences] belong here. Several important discoveries of this field are [Fermat's little theorem] , [Euler's theorem] , the [Chinese remainder theorem] and the law of [quadratic reciprocity] . The properties of [multiplicative function] s such as the [Möbius function] and [Euler's φ function] , [integer sequence] s, [factorial] s, and [Fibonacci number] s all also fall into this area.

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