Q1.
The next two taxicab numbers after 1729 are 4104 and 13832. Find the two ways in which each of these can be expressed as the sum of two positive cubes.
Answer
Search the cubes below each number and look for two that add up.
4104 = 23 + 163 = 8 + 4096
4104 = 93 + 153 = 729 + 3375
4104 = 93 + 153 = 729 + 3375
13832 = 23 + 243 = 8 + 13824
13832 = 183 + 203 = 5832 + 8000
13832 = 183 + 203 = 5832 + 8000
How to search efficiently: For 4104, the largest possible cube is 163 = 4096, since 173 = 4913 is already too big. So run a from 1 to 16, subtract a3 from 4104, and check whether what is left is a cube. Only a = 2 and a = 9 work. The same sweep on 13832 with a up to 24 gives a = 2 and a = 18.
Did you know? Hardy's remark that 1729 was "rather a dull number" is what drew Ramanujan's reply that it is the smallest number expressible as a sum of two cubes in two different ways: 1729 = 13 + 123 = 93 + 103. Notice that 4104 = 23 × 513 and 13832 = 23 × 1729 — the third taxicab number is just 8 times the first.