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.

Formulae yielding prime numbers

Main article formula for primes

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There is no formula for primes which is more efficient at finding primes than the methods mentioned above under "Finding prime numbers". Those which do exist have little practical value.

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The curious polynomial f(n) = n2 − n + 41 yields primes for n = 0,..., 40, but f(41) is composite. It has been proved that there is no polynomial which only yields prime numbers in this fashion.

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There is a set of Diophantine equations in 9 variables and one parameter with the following property: the parameter is prime if and only if the resulting system of equations has a solution over the natural numbers. This can be used to obtain a single formula with the property that all its positive values are prime.

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Another formula is based on Wilson's theorem mentioned above, and generates the number two many times and all other primes exactly once. There are other similar formulae which also produce primes.

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