8191

This number is a prime.

+ 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+19-181+191+199-811... 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 3.0-XP, 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 13-step number. Note that 2^13-1 = 8191, making it a Mersenne prime also. Will Mersenne primes continue to appear in this way? [Loungrides and Honaker]

+ 8191 is a Mersenne prime (2^13-1) that is the sum of 13 consecutive primes, from 599 to 661. [Rivera]

+ The first comment from Bateman comes from the Goormaghtigh conjecture: the only two non-trivial integer solutions of the exponential Diophantine equation (x^m-1)/(x-1) = (y^n-1)/(y-1) with x > y > 1 and m,n > 2 give primes 31 and 8191, with here 8191 = (90^3-1)/89 = (2^13-1)/1 and 8191 = 111_90 = 1111111111111_2. So, 8191 is one of the only two primes to be Brazilian in two distinct bases, the second is 31. [Schott]

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