# 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.

(There are 10 curios for this number that have not yet been approved by an editor.)