One thousand, seven hundred and twenty nine.
The story goes that the English professor of mathematics G. H. Hardy was once traveling to see his student Srinivasa Ramanujan. As Hardy relates:
I remember once going to see him while he was lying ill at Putney. I had ridden in taxi-cab No. 1729, and remarked that the number seemed to me rather a dull one, and I hoped it was not an unfovourable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as a sum of two cubes in two different ways."
The story is used as an example of the remarkable mental powers possessed by Ramanujan.
This question cropped up once on University Challenge: "What is the smallest number expressible as a sum of two cubes in two different ways?" Naturally, some smartypants immediately buzzed and answered 1729, treating the delighted viewer to probably the first time in history that Jeremy Paxman had been rendered speechless. As I recall his eventual retort was along the lines of, "bloo... dy... hell".