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.

Prime gaps

Let pn denote the n-th prime number (i.e. p1 = 2, p2 = 3, etc.). The gap gn between the consecutive primes pn and pn + 1 is the number of (composite) numbers between them, i.e.

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:gn = pn + 1 − pn − 1.

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(Slightly different definitions are sometimes used.) We have g1 = 0, g2 = g3 = 1, and g4 = 3. The sequence {gn} of prime gaps has been extensively studied.

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For any N, the sequence

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:(N + 1)! + 2, (N + 1)! + 3, ..., (N + 1)! + N + 1

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is a sequence of N consecutive composite integers. Therefore, there exist gaps between primes which are arbitrarily large, i.e. for any natural number N, there is an integer n with gn > N. (Choose n so that pn is the greatest prime number less than (N + 1)! + 2.) On the other hand, the gaps get arbitrarily small in proportion to the primes: the quotient (gn/pn) approaches zero as n approaches infinity.

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We say that gn is a maximal gap if gm < gn for all m < n. The largest known maximal gap is 1131, found by T. Nicely and B. Nyman in 1999. It is the 64th smallest maximal gap, and it occurs after the prime 1693182318746371.

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The largest prime gap with identified gap ends known as of 1 January 2005 has a

Related Topics:
1 January - 2005

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length of 2254930 http://hjem.get2net.dk/jka/math/primegaps/megagap2.htm.

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Note that the Twin Prime Conjecture simply asserts that gn = 1 for infinitely many integers n.

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