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taxicab numbers

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  • Brackets around 1729, please. :-)

    March 7, 2010

  • The nth taxicab number Ta(n) is the smallest number representable in n ways as a sum of positive cubes.

    The name is derived from the second taxicab number, Ta(2) = 1729, which can be represented as both the sum of 10 cubed and 9 cubed and the sum of 12 cubed and 1 cubed. Ta(2), also known as the Hardy-Ramanujan number, achieved immortality following an incident where Hardy visited Ramanujan in hospital. According to Hardy:

    I remember once going to see him when he was lying ill at Putney. I had ridden in taxi-cab No. 1729, and remarked that the number seemed to be rather a dull one, and that I hoped it was not an unfavourable 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."

    March 7, 2010