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GIMPS has discovered a new largest known prime number: 2^{82589933}1 (24,862,048 digits) In "A Density Version of the Vinogradov Three Prime Theorem" Xuancheng Shao proves that if A is a subset of the primes, and the lower density of A in the primes is larger than 5/8, then all sufficiently large odd positive integers can be written as the sum of three primes in A. The constant 5/8 in this statement is the best possible. [Post]
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