1729 is known as the Hardy–Ramanujan number after a famous anecdote of the British mathematician G. H. Hardy regarding a visit to the hospital to see the Indian mathematician Srinivasa Ramanujan. In Hardy’s words:

I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. “No,” he replied, “it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.”

The two different ways are these:

1729 = 1^{3} + 12^{3} = 9^{3} + 10^{3}

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):

91 = 6^{3} + (−5)^{3} = 4^{3} + 3^{3}

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.

I'm a Professor of Mathematics and a University Distinguished Teaching Professor at the University of North Texas. For eight years, I was co-director of Teach North Texas, UNT's program for preparing secondary teachers of mathematics and science.
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I have a copy of Newton’s Principles of Natural Philosophy (3rd English edition) Dated 1729 !

I have a copy of Newton’s Principles of Natural Philosophy (3rd English edition) Dated 1729 !