The Hardy-Ramanujan number $1729$ is the smallest number expressible as

The Hardy-Ramanujan number $1729$ is the smallest number expressible as
  1. the sum of two cubes in two different ways
  2. the cube of a prime
  3. a perfect square
  4. the sum of three squares

Solution

$1729 = 1^{3} + 12^{3} = 9^{3} + 10^{3}$ — the famous taxicab number.

Asked in: IMO

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