# 1729

The Hardy-Ramanujan number is the smallest product of three distinct primes of the form 6*n* + 1. [Pol]

The largest number which is divisible by its prime sum of digits (19) and reversal (91)
happens to be Ramanujan's famous taxi-cab number
(1729 = 12^{3} + 1^{3} = 10^{3} + 9^{3}).
It is the smallest number expressible as the sum of two positive cubes in two different ways.

The smallest number that is a pseudoprime simultaneously to bases 2, 3 and 5. [Pomerance , Selfridge and Wagstaff]

If you reverse the middle digits of this pseudoprime you get 1279 and 2^{1279} - 1 is a Mersenne prime. [Luhn]

Schiemann's first pair of isospectral lattices L^{+}(1,7,13,19) and L^{-}(1,7,13,19) are of determinant 1*7*13*19 = 1729. [Poo Sung]

The Hardy-Ramanujan number is equal to the average of the only known prime squares of the form n! + 1, i.e., 25, 121, and 5041. [Gudipati]