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.

How many prime numbers are there?

There are infinitely many prime numbers. The oldest known proof for this statement is given by the Greek mathematician Euclid in his Elements (Book IX, Proposition 20). Euclid states the result as "there are more than any given number of primes", and his proof is essentially the following:

Related Topics:
Infinite - Euclid

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:Suppose you have a finite number of primes. Call this number m. Multiply all m primes together and add one (see Euclid number). The resulting number is not divisible by any of the finite set of primes, because dividing by any of these would give a remainder of one. Therefore it must either be prime itself, or be divisible by some other prime that was not included in the finite set. Either way, there must be at least m+1 primes. But this argument applies no matter what m is; it applies to m+1, too. So there are more primes than any given finite number.

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Other mathematicians have given their own proofs. One of those (due to Euler) shows that the sum of the reciprocals of all prime numbers diverges to infinity. Kummer's is particularly elegant and Furstenberg provides one using general topology.

Related Topics:
Euler - The sum of the reciprocals of all prime numbers diverges to infinity - Kummer - General topology

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Even though the total number of primes is infinite, one could still ask "approximately how many primes are there below 100,000" or "How likely is a random 100-digit number to be prime?" Questions like these are answered by the prime number theorem.

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