# 8191

This number is a prime.

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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]
Submitted: 2019-09-25 07:39:14; Last Modified: 2020-01-30 06:40:21.

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