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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 ...
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.
Legendre's conjecture, proposed by Adrien-Marie Legendre, states that there is a prime number between and for every positive integer . The conjecture is one of Landau's problems (1912) on prime numbers, and is one of many open problems on the spacing of prime numbers. Unsolved problem in mathematics: Does there always exist at least one 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 .
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 ...
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 ...
It suffices to check that each prime gap starting at p is smaller than . A table of maximal prime gaps shows that the conjecture holds to 2 64 ≈ 1.8 × 10 19. [21] A counterexample near that size would require a prime gap a hundred million times the size of the average gap.
Polignac's conjecture. In number theory, Polignac's conjecture was made by Alphonse de Polignac in 1849 and states: [1] For any positive even number n, there are infinitely many prime gaps of size n. In other words: There are infinitely many cases of two consecutive prime numbers with difference n.