Why is 1729 called the Ramanujan number (or Hardy-Ramanujan number)?

The number 1729 is famously known as the Ramanujan number or Hardy-Ramanujan number due to an anecdote involving the British mathematician G.H. Hardy and the Indian mathematical genius Srinivasa Ramanujan.

The Anecdote: As told by G.H. Hardy, he once visited Ramanujan when Ramanujan was ill in Putney (London). Hardy remarked that he had ridden in a taxi-cab with the number 1729, which he thought was "a rather dull number." Ramanujan immediately replied, "No, it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."

The Mathematical Property:
1729 = 1³ + 12³ = (1×1×1) + (12×12×12) = 1 + 1728 = 1729
1729 = 9³ + 10³ = (9×9×9) + (10×10×10) = 729 + 1000 = 1729

Significance: This story beautifully illustrates Ramanujan's extraordinary intuition and familiarity with numbers. That he could recall such a specific property of a seemingly random number on the spot amazed Hardy and has become a classic example of Ramanujan's genius.

Taxicab Numbers: Numbers that are the smallest number expressible as the sum of two cubes in n different ways are now sometimes called "taxicab numbers." 1729 is the second taxicab number, Ta(2). (Ta(1) = 2 = 1³ + 1³).

Key Takeaway: 1729 is called the Ramanujan number because of a famous story where Srinivasa Ramanujan immediately recognized it as the smallest positive integer that can be expressed as the sum of two cubes in two different ways, showcasing his remarkable number sense.