1729

1729 is a natural number that follows 1728 and precedes 1730. In number theory, it is notable for two distinct representations as a sum of two cubes: 1729 = 1^3 + 12^3 = 9^3 + 10^3. This makes it the smallest positive integer expressible as the sum of two cubes in two different ways, a property that has earned it the nickname the Hardy–Ramanujan number and the designation Taxicab(2).

Aside from its role in additive number theory, 1729 has a prime factorization of 7 × 13 × 19. It is also a Carmichael number, meaning it is composite and satisfies a^n ≡ a (mod n) for every integer a that is relatively prime to n. For Carmichael numbers, each prime factor p satisfies p−1 dividing n−1; indeed, 6, 12, and 18 divide 1728 in this case.

Historically, the number is associated with a famous episode involving G. H. Hardy and Srinivasa Ramanujan. When Hardy visited Ramanujan in a hospital, he remarked that 1729 seemed to be a dull number, to which Ramanujan reportedly replied that it was very interesting precisely because of its two distinct cube-sum representations.

1729 is sometimes cited in discussions of taxicab numbers and Carmichael numbers and is used as an example in recreational and theoretical number theory. It also appears in various expository treatments of Diophantine equations and integer relations.