1729
This number is a composite.
The Hardy-Ramanujan number is the smallest product of three distinct primes of the form 6n + 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 = 123 + 13 = 103 + 93).
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 21279 - 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]
1729^17+1729^29-1 is a Pythagorean prime of the form
(4n+1). [Bajpai]
The smallest Carmichael number of the form
(6*k+1)*(12*k+1)*(18*k+1), where 6*k+1, 12*k+1 and 18*k+1.