The first prime Lychrel number (a number that does not form a palindrome by repeatedly applying the reverse-then-add process). The first Lychrel number is the reversal of 691.
G. N. Watson proved that the Ramanujan tau function is divisible by 691 for almost all positive integers. [Terr]
The first irregular prime to appear in the numerator of a Bernoulli number.
The only known prime number which is a square when turned upside down (169) and another square when reversed (196). Note that 169 and 196 are the smallest consecutive squares using the same digits. [Capelle]
The smallest prime that can be written as the sum of thirteen consecutive primes. Recall that 691 is thirteen squared, turned upside down. [Post]
The number of prime unlabeled topologies (i.e., prime homeomorphism classes) on 6 points, and the largest known prime of that form. [Post]
The smallest prime coefficient in the the minimal
resolution of the Sylow 2-subgroup of the Mathieu group M24
out to degree 13: the Poincare series being 1 + 4t + 12t^2
+ 25t^3 + 49t^4 + 85t^5 + 143t^6 + 222t^7 + 336t^8 + 485t^9
+ 691t^10 + 953t^11 + 1298t^12 + 1726t^13 + O(t^14). [Post]
There are exactly 691 water fountains in the Pentagon. [Hawkins]
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