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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+19-181+191+199-811... the sum is 1381,
which is a prime as well. [Meiburg]
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