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A prime gap is the difference between two successive prime numbers. The n -th prime gap, denoted gn or g ( pn) is the difference between the ( n + 1)-st and the n -th prime numbers, i.e. We have g1 = 1, g2 = g3 = 2, and g4 = 4. The sequence ( gn) of prime gaps has been extensively studied; however, many questions and conjectures remain unanswered.
For example, among the positive integers of at most 1000 digits, about one in 2300 is prime (log(10 1000) ≈ 2302.6), whereas among positive integers of at most 2000 digits, about one in 4600 is prime (log(10 2000) ≈ 4605.2). In other words, the average gap between consecutive prime numbers among the first N integers is roughly log(N). [3]
A simple formula is. for positive integer , where is the floor function, which rounds down to the nearest integer. By Wilson's theorem, is prime if and only if . Thus, when is prime, the first factor in the product becomes one, and the formula produces the prime number . But when is not prime, the first factor becomes zero and the formula ...
Twin prime. A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (17, 19) or (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term twin prime is used for a pair of twin primes; an alternative name for this is prime ...
A prime number (or prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. By Euclid's theorem, there are an infinite number of prime numbers. Subsets of the prime numbers may be generated with various formulas for primes.
Cramér's conjecture. In number theory, Cramér's conjecture, formulated by the Swedish mathematician Harald Cramér in 1936, [1] is an estimate for the size of gaps between consecutive prime numbers: intuitively, that gaps between consecutive primes are always small, and the conjecture quantifies asymptotically just how small they must be.
In number theory, primes in arithmetic progression are any sequence of at least three prime numbers that are consecutive terms in an arithmetic progression. An example is the sequence of primes (3, 7, 11), which is given by for . According to the Green–Tao theorem, there exist arbitrarily long arithmetic progressions in the sequence of primes.
Bertrand's postulate was proposed for applications to permutation groups. Sylvester (1814–1897) generalized the weaker statement with the statement: the product of k consecutive integers greater than k is divisible by a prime greater than k. Bertrand's (weaker) postulate follows from this by taking k = n, and considering the k numbers n + 1 ...