1,729

In 1729, James Bradley explained the aberration of starlight, the first direct evidence that the Earth moves (England).

1729 is known as the Hardy–Ramanujan number after G. H. Hardy's published account of remarking that his taxicab's number seemed dull, to which Srinivasa Ramanujan replied that it was the smallest number expressible as a sum of two cubes in two different ways.

1729 = 7 × 13 × 19 is composite and still passes Fermat's primality test for every base coprime to it, which makes it the third Carmichael number after 561 and 1105. This is arguably a more consequential property than the taxicab one, since Carmichael numbers are the reason practical primality testing must be probabilistic and repeated — and it is almost never the fact quoted, because the anecdote is better.

1729 is Ta(2), the smallest number expressible as a sum of two positive cubes in two ways. The sequence continues but very steeply: Ta(3) is 87,539,319, found in 1957, and only six terms are known at all. Ta(7) and beyond have never been determined. The gap between the first term, which fits on a taxi plate, and the second, which has eight digits, is why the anecdote is memorable and the sequence is not.

1729 is the smallest number expressible as the sum of two positive cubes in two distinct ways: 1³ + 12³ = 9³ + 10³ = 1729. It is the first of the 'taxicab numbers'.

1729 recurs in The Simpsons and Futurama, whose writing staff included several mathematics graduates: it appears as a starship registry number, a bus number and elsewhere, planted deliberately as a reference to the Hardy–Ramanujan story. The pattern is documented by the writers themselves rather than inferred by viewers, which is what separates it from the ordinary noise of numbers appearing in television.