# 691

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]