Ramanujan can do this instantly

Sunday, December 22 will be Srinivasa Ramanujan's 126126th birthday. Ramanujan was an amazing mathematician, but one of the things he was most famous for had to do with the number 17291729. When Ramanujan was in the hospital, he was visited by his friend G.H. Hardy. Hardy remarked that the taxicab that he had ridden in had a rather uninteresting number: 17291729. Ramanujan said that no, 17291729 was very interesting because it was the smallest number that can be expressed as the sum of 22 cubes in 22 different ways. These are 123+1312^3+1^3 and 103+9310^3+9^3. Hence, numbers that can be written as the sum of multiple cubes are called taxicab numbers.

In the spirit of the sums of cubes, 126126 can be written as the sum of two positive cubes, A3A^3 and B3B^3. What is A+B A + B ?

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