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The smallest number that can be written as the sum of two cubes in two different ways is
Which pair of cubes gives 1729 in the second way (other than 1^3 + 12^3)?
4104 can be written as the sum of two cubes in two different ways.
Show that 20683 = 10^3 + 27^3 = 19^3 + 24^3. What special name is given to such numbers?
(a) Show that 39312 = 2^3 + 34^3 = 15^3 + 33^3. (b) Using prime factorisation, show that 39312 itself is not a perfect cube.
The mathematician G. H. Hardy once visited S. Ramanujan in hospital and remarked that his taxi number, 1729, seemed rather dull. Ramanujan replied that it was very interesting: it is the smallest number that can be written as the sum of two cubes in two different ways. Such numbers are now called Hardy–Ramanujan numbers.
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