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1 + 2 + 2^2 + 2^3 + ... + 2^12 = 1 + 90 + 90^2 = 8191. [Bateman] This number turned upside down forms the first four digits in the decimal expansion of the golden ratio (phi = 1.618...). [Wu] There is only one prime less than 8191 that is also a repunit in three bases. Can you find it? [Pimentel] The smallest Mersenne prime p such that the Mersenne number M(p) = 2^p  1 is composite. The smallest Mersenne prime corresponding to an emirp (M(13) = 8191). [Loungrides] All Mersenne primes are of the from 2^k  1, and k must be prime. For 8191, k=13. The case of k=11 (the prime preceding 13) gives the smallest composite Mersenne number, with one of it's factors being 89. If you remove every other digit, you get 89 as well. [Meiburg] Suppose you listed all primes, with three digits or less, using the digits in this number (1,8, and 9), without limiting the number of times a number can use each digit (i.e., it may have two nines). If they are now listed from least to greatest with signs in the order "+, +, ,", beginning with 11+18+19181+191+199811... the sum is 1381, which is a prime as well. [Meiburg] 8191 Mersenne (1993 OX9) is an asteroid discovered on 20 July 1993, by Eric Walter Elst at La Silla Observatory.
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