1729 is the natural number following 1728 and preceding 1730. It is known as the Hardy–Ramanujan number, after an anecdote of the British mathematician G. H. Hardy when he visited Indian mathematician Srinivasa Ramanujan in hospital. He related their conversation:

The two different ways are:

The quotation is sometimes expressed using the term "positive cubes", since allowing negative perfect cubes (the cube of a negative integer) gives the smallest solution as 91 (which is a divisor of 1729):

Numbers that are the smallest number that can be expressed as the sum of two cubes in n distinct ways have been dubbed "taxicab numbers". The number was also found in one of Ramanujan's notebooks dated years before the incident, and was noted by Frénicle de Bessy in 1657.

The same expression defines 1729 as the first in the sequence of "Fermat near misses" (sequence in the OEIS) defined as numbers of the form 1 + z³ which are also expressible as the sum of two other cubes.

1729 is also the third Carmichael number and the first absolute Euler pseudoprime. It is also a sphenic number.

1729 is a Zeisel number. It is a centered cube number, as well as a dodecagonal number, a 24-gonal and 84-gonal number.

Investigating pairs of distinct integer-valued quadratic forms that represent every integer the same number of times, Schiemann found that such quadratic forms must be in four or more variables, and the least possible discriminant of a four-variable pair is 1729 (Guy 2004).

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