Question

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.