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GIMPS has discovered a new largest known prime number: 2^{82589933}1 (24,862,048 digits) 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. Each term in the sequence of primes 7, 89, 5591, 3851459, ..., has a Mersenne prime number of consecutive composites that follow them. Can you find a prime that is followed by a gap of 8191 consecutive composites? (Someday, someone will find its first occurrence!) [Honaker] In the version of Solitaire that came with Windows 3.0XP, if playing in the timed mode, the timer stops at 8191 and will not continue to count up at that point. [Forest] 8191 is a Fibonacci 13step number. Note that 2^131 = 8191, making it a Mersenne prime also. Will Mersenne primes continue to appear in this way? [Loungrides and Honaker]
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