Prime number

In mathematics, a prime number (or prime) is a natural number greater than one whose only positive divisors are one and itself. Or for short: A prime number is a natural number with exactly two natural divisors. A natural number that is greater than one and is not a prime is called a composite number. The numbers zero and one are neither prime nor composite. The property of being a prime is called primality. Prime numbers are of fundamental importance in number theory.

Open questions

There are many open questions about prime numbers. The most significant of these is the Riemann hypothesis, which essentially says that the primes are as regularly distributed as possible. From a physical viewpoint, it roughly states that the irregularity in the distribution of primes only comes from random noise. From a mathematical viewpoint, it roughly states that the asymptotic distribution of primes (about 1/ log x of number less than x are primes, the prime number theorem) also holds for much shorter intervals of length about the square root of x (for intervals near x). This hypothesis is generally believed to be correct, in particular, the simplest assumption is that primes should have no significant irregularities without good reason.

Related Topics:
Riemann hypothesis - Prime number theorem

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Other famous conjectures have a much greater chance of being true (in a formal sense, they follow from simple heuristic probabilistic arguments) with the lack of a solution more of a reflection of lack of good technical tools (so theoretical physicists would just regard them as being true):

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• Goldbach's conjecture: Can every even integer greater than 2 be written as a sum of two primes?
• Twin prime conjecture: A twin prime is a pair of primes with difference 2, such as 11 and 13. Are there infinitely many twin primes?
• Does the Fibonacci sequence contain an infinite number of primes?
• Are there infinitely many Mersenne primes and Fermat primes? The expected answers are yes, resp. no.
• Are there infinitely many primes of the form n² + 1?
• Legendre's conjecture: Is there always a prime number between n² and (n + 1)² for every n?
• Cramer's conjecture that $d(x)$, the largest gap between consecutive primes, among all primes less than x, is asymptotic to $log^2\; x$. This conjecture clearly implies the previous one, but its status is a little more unsure.
• Brocard's conjecture: Are there always at least four primes between the squares of successive primes > 2?