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GIMPS has discovered a new largest known prime number: 2^{82589933}1 (24,862,048 digits) Bertrand's postulate implies that there are infinitely many numbers a for which floor(2^{a}), floor(2^{2a}), floor(2^{22a}), ... are all prime. The smallest of these is a = 1.25164 75977 90463 01759 44320 53623... and generates the Bertrand primes: 2, 5, 37, 137438953481, .... The next Bertrand prime has 41,373,247,571 digits. [Caldwell]
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