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Number theory

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Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers 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 term "arithmetic" is also used to refer to number theory. This is a somewhat older term, which is no longer as popular as it once was. Number theory used to be called the higher arithmetic, but this too is dropping out of use. Nevertheless, it still shows up in the names of mathematical fields (arithmetic functions, arithmetic of elliptic curves, fundamental theorem of arithmetic). This sense of the term arithmetic should not be confused either with elementary arithmetic, or with the branch of logic which studies Peano arithmetic as a formal system. Mathematicians working in the field of number theory are called number theorists.

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In mathematics, the Pell numbers and companion Pell numbers are both sequences of integers that have been known since ancient times. They are defined by a recurrence relation similar to that for the Fibonacci numbers, and grow exponentially, proportionally to powers of the silver ratio. Pell numbers arise in the approximation of the square root of 2, in the definition of square triangular numbers, in the construction of nearly-isosceles integer right triangles, and in certain combinatorial enumeration problems.

As with Pell's equation, the name of the Pell numbers stems from Leonhard Euler's mistaken attribution of the equation and the numbers derived from it to John Pell. The Pell-Lucas numbers are also named after Edouard Lucas, who studied sequences defined by recurrences of this type; the Pell and companion Pell numbers are Lucas sequences.


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Euclidean algorithm running time

Plot of the running time of the Euclidean algorithm for gcd(x,y). Red indicates a fast computation, while successively bluer points indicate slower computations.

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Algebraic number theory • Analytic number theory • Arithmetic functions • Computational number theory • Continued fractions • Diophantine approximation • Diophantine equations • Elementary number theory • Geometry of numbers • Integer sequences • Lattice points • Modular arithmetic • Modular forms • Multiplicative functions • Number theoretic algorithms • Number theorists • Prime numbers • Quadratic forms
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Topics in Number theory

Types of number theory Numbers Equations Arithmetic
  • Numbers
  • Natural numbers
  • Prime numbers
  • Rational numbers
  • Irrational number
  • Transcendental number
  • Quadratic forms
  • L-functions
  • Diophantine equations
  • Diophantine approximation
  • Continued fraction
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