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Taxicab numbers

Taxicab numbers

Once, in a taxi (in English taxicab) in London, Hardy was struck by his car number, in 1729. He must have been thinking about it because he entered the hospital room where Ramanujan was lying in bed and With a dry "hello," he expressed disappointment about this number. It was, according to him, a boring number, adding that he hoped it was not a bad omen.

No, Hardy, Ramanujan said, is a very interesting number. It is the smallest number expressible as the sum of two positive cubes in two different ways.

What are the two pairs of numbers that are raised to the cube and added together give us 1729?

Solution

1729 can be expressed as 13 + 123, or 93 + 103.

This is a rather strange property, only the so-called “taxicab numbers” share it, of which only the first 5 members of the list are known: 2, 1729, 87539319, 6963472309248 and 48988659276962496.