62Number Theory

The number theory is a branch of mathematics dealing with the divisibility properties of integers with a special focus on primes. Primes are natural numbers, which are divisible only by 1 and by themselves.

Many problems of number theory can be easily formulated, for instance:

1. What is the number of divisors of a given natural number?
2. What is the greatest common divisor of two given natural numbers?
3. Are there infinitely many primes?
4. Can the sum \(\sum_{n=1}^\infty\frac{1}{n}\) be calculated and, if yes, what is its result?
5. Can the sum \(\sum_{p\text{ prime}}^\infty\frac{1}{p}\) be calculated and, if yes, what is its result?
6. Can the number of primes between a given \(n\) and \(2n\) be estimated in terms of \(n\)?

This simple description of number theory might create the impression that it is a very elementary discipline of mathematics, since it deals with the most common numbers known from the high school and its problems can be easily stated. However, this impression is wrong. Surprisingly, over centuries, many great mathematicians had first to develop some sophisticated instruments and concepts in order to reveal the secrets of natural numbers and primes.

This process has continued to the present and there are still many number-theoretic open problems and conjectures waiting for being proved or disproved.

| | | | Contributors: bookofproofs

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763.Algebraic Number Theory

4274.Additive Number Theory