# Portal:Number theory

## 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**.

## Selected article

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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## Selected picture

Graph of the number of ways to write an even number n as the sum of two primes (4 ≤ n ≤ 1,000,000). This is the main object of study of the Goldbach's conjecture

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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 only 35 even numbers have been identified which are not the sum of a pair of Twin primes?
- ...that 16 is the only integer that satisfies x^y=y^x for distinct positive integers x and y, being 4^2 = 2^4

## Categories

*Topics in Number theory*

Types of number theory | Numbers | Equations | Arithmetic |
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