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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 ...
Green–Tao theorem. In number theory, the Green–Tao theorem, proved by Ben Green and Terence Tao in 2004, states that the sequence of prime numbers contains arbitrarily long arithmetic progressions. In other words, for every natural number k, there exist arithmetic progressions of primes with k terms. The proof is an extension of Szemerédi ...
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.
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.
All prime numbers from 31 to 5,131,135,981 for free download. Lists of Primes at the Prime Pages. The Nth Prime Page Nth prime through n=10^12, pi(x) through x=3*10^13, Random prime in same range. Prime Numbers List Full list for prime numbers below 10,000,000,000, partial list for up to 400 digits.
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 ...
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Andrica's conjecture (named after Romanian mathematician Dorin Andrica) is a conjecture regarding the gaps between prime numbers. [1] The conjecture states that the inequality. holds for all , where is the n th prime number. If denotes the n th prime gap, then Andrica's conjecture can also be rewritten as.