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GIMPS has discovered a new largest known prime number: 2^{82589933}1 (24,862,048 digits) If you pick two integers at random, the probability that they are relatively primes is 6/^{2}. [Caldwell] 1/zeta(2) where zeta is the Riemann zeta function = 0.6079271018540266286... = the asymptotic density of squarefree numbers (also called quadratfrei), those whose prime decomposition contains no repeated factors. All primes are therefore trivially squarefree. The number 1 is by convention taken to be squarefree. The squarefree numbers are 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, .... Conversely, the squareful numbers (i.e., those that contain at least one square) are 4, 8, 9, 12, 16, 18, 20, 24, 25, .... [Post]
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