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GIMPS has discovered a new largest known prime number: 2^{82589933}1 (24,862,048 digits) The largest known Fermat prime (2^{24} + 1). Just a small proportion of regular polygons (ngons) can be constructed with compass and straightedge. Gauss proved that if n is a Fermat prime, then it is possible to construct an ngon. Wantzel later proved this condition was also necessary (for prime ngons), so the 65537gon is currently the largest known constructible prime ngon. It took Hermes 10 years and a 200page manuscript to write down a procedure for its construction. Would you like to attempt it? The smallest prime that is the sum of a nonzero square and a nonzero cube in four different ways: 65537 = 122^{2} + 37^{3} = 219^{2} + 26^{3} = 255^{2} + 8^{3} = 256^{2} + 1^{3}. [Post] To remember the digits of 65537, recite the following mnemonic: "Fermat prime, maybe the largest." Then count the number of letters in each word. [Brent] Largest known prime mean of a Fermat prime and a Mersenne prime 65537 = (3+131071)/2. Richard Mathar has searched through all means that can be created from the existing values of the two OEIS sequences, Mersenne primes and Fermat primes. [Post] The largest nontitanic prime of form n^(2*n) + 1, (case n = 4). [Loungrides] The smallest octavan prime (form x^8 + y^8), where x and y are distinct nonprime digits, i.e., 1^8 + 4^8 = 65537. [Loungrides] The largest known prime with a Hamming Weight of 2. In other words, the binary representation of 65537 has only 2 ones, which makes computer operations much easier. This fact is extremely important to cryptographers, who use 65537 as a common encryption exponent. [Price]
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