Numbers which are the smallest number expressible as the sum of two positive cubes in n distinct ways are called taxicab numbers.
This quite peculiar name is a result of a story, probably a true story, involving mathematicians G. H. Hardy and the great Srinivasa Ramanujan.
Hardy describes when he visited Ramanujan in India:
"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 unfavourable omen. ‘No,’ [Ramanujan] 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 cubes are 1729 = 1^3 + 12^3 = 9^3 + 10^3.
This conversation, taking place in a taxi, gave the name to taxicab numbers, and 1729 is called The Hardy-Ramanujan Number.
Srinivasa Ramanujan (1878-1920) was one of the most remarkable mathematicians in history. Lacking formal training in higher mathematics, he developed his research in isolation with astonishing results. Established mathematicians ignored him though; his results were too remarkable (including solutions to problems so far considered unsolvable) and he did not present them in the conventional way.
In 1913, he managed to get in touch with G.H. Hardy, a mathematician at the University of Cambridge. Hardy saw the greatness of Ramanujan and they remained friends until Ramanujan's death in 1920.
The sum of Ramanjuan's work is impressing. He opened up for completely new areas of mathematics. Still 100 years after his death, new discoveries were made in his notebooks.
(The lead image shows a detail of special case integrals for constants defined by Ramanujan's theta function.)
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You find more articles about different aspects of numbers in Community Numbers (7b6a).
Interesting. I have never read about taxicab numbers before. I actually find prime numbers particularly fascinating. These cannot be calculated with any multiplications.