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Bertrand's postulate
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== Sylvester's theorem == Bertrand's postulate was proposed for applications to [[permutation group]]s. [[James Joseph Sylvester|Sylvester]] (1814β1897) 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, ''n'' + 2, up to and including ''n'' + ''k'' = 2''n'', where ''n'' > 1. According to Sylvester's generalization, one of these numbers has a prime factor greater than ''k''. Since all these numbers are less than 2(''k'' + 1), the number with a prime factor greater than ''k'' has only one prime factor, and thus is a prime. Note that 2''n'' is not prime, and thus indeed we now know there exists a prime ''p'' with ''n'' < ''p'' < 2''n''.
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