Portal:Number theory

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Number theory

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, arithmetic geometry). 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 number theory, Znám's problem asks which sets of k integers have the property that each integer in the set is a proper divisor of the product of the other integers in the set, plus 1. Znám's problem is named after the Slovak mathematician Štefan Znám, who suggested it in 1972, although other mathematicians had considered similar problems around the same time. One closely related problem drops the assumption of properness of the divisor, and will be called the improper Znám problem hereafter.

One solution to the improper Znám problem is easily provided for any k: the first k terms of Sylvester's sequence have the required property. Sun (1983) showed that there is at least one solution to the (proper) Znám problem for each k ≥ 5. Sun's solution is based on a recurrence similar to that for Sylvester's sequence, but with a different set of initial values.

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Time-keeping on a clock gives an example of modular arithmetic, the "clock group" is represented by the group Z/12Z for a 12-hour clock and Z/24Z for a 24-hour clock.

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Did you know?

• ...that every positive integer can be expressed as the sum of four squares of integers?
• ...that it is impossible to separate any power higher than the second into two like powers?
• ...that the only integer that can be written as reverse powers is 16, i.e. 16 = 2^4 = 4^2?

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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
Algebraic number theory Analytic number theory Geometric number theory Computational number theory Transcendental number theory	Numbers Natural numbers Prime numbers Rational numbers Irrational number Transcendental number	Quadratic forms L-functions Diophantine equations Diophantine approximation Continued fraction	Arithmetic Modular arithmetic Arithmetic functions Fundamental theorem of arithmetic

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