This number is a prime.

+ The first prime Lychrel number. Note that 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]

+ The largest known prime p such that tau(p) is congruent to 1 (mod p), where tau is Ramanujan's tau function. [Luen]

+ The sum of the first 691 composite numbers is divisible by 666. [Gaydos]

+ 691 = 6!-6#+1. [Galliani]

+ The Dome of the Rock was completed in 691 CE.

+ 691 = prime(125) = (12 squared + 5 squared) turned upside down. [Worrom]

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