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GIMPS has discovered a new largest known prime number: 2^{82589933}1 (24,862,048 digits) The generalized repunit (675065^{1153}  1)/675064 is prime. A proof is possible because of the remarkable circumstance that, for b = 5*37*41*89 and p = 2^{7}*3^{2}+1, both Phi(p,b) and Phi(p1, b) are prime, where Phi(n, x) is the n^{th} cyclotomic polynomial in x. [Broadhurst]
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