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GIMPS has discovered a new largest known prime number: 2^{82589933}1 (24,862,048 digits) The 46th prime is 199. Let C = the number of combinations of a group of 199 objects taken 46 at a time. Note that 199!/[46!(19946)!] has 46 digits and C ± 1 are both primes! [Hartley] The least even number 2k such that SOP(2, P(k)) is divisible by 2k, i.e., SOP(2, P(k)) is congruent to 0 mod (2k). Note that SOP(2, P(k)) = the sum of primes from 2 to the prime P(k). [Rivera] Mersenne wrongly thought 2^671 was prime. 46 is the smallest base b for which (b^671)/(b1) is actually prime.
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