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Ramanujan was a genius, but he didn't just factor 1729 on the spot in the hospital bed. According to Wikipedia, he'd already known about it a few years before,

• justifier 11y ago
  when you think about these sorts of things you come across many interesting numbers that posit unique problems to your theories when you are looking to develop a theory you can sometimes look to markers for guidance markers like: first, smallest, largest known, outliers; ramanujan's language: the smallest number expressible as a sum of two cubes in two different ways. certainly implies he once asked himself.. what is the smallest number i can express as the sum of two cubes in only two different ways and its place as the smallest, with postive cubes, means it is elementary and essential to a number of extended applications that use products,cubes and sums i wish the story went on to have ramanujan explain what he was seeking to uncover by examining which number could be the smallest expressible as the sum of two cubes in two different ways.. to what end was he using the two different ways, controlled in their only being two, to examine properties of sums of cubes