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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, 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, the prime number theorem describes the asymptotic distribution of the prime numbers. The prime number theorem gives a rough description of how the primes are distributed.

Roughly speaking, the prime number theorem (PNT) states that if you randomly select a number nearby some large number N, the chance of it being prime is about 1 / ln(N), where ln(N) denotes the natural logarithm of N. For example, near N = 10,000, about one in nine numbers is prime, whereas near N = 1,000,000,000, only one in every 21 numbers is prime.

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