1729

There’s a famous story in the math world about seemingly “boring” numbers. It goes like this:

There was a famous Indian mathematician named Srinivasa Ramanujan. Mr Ramanujan went to England to study in England with another famous mathematician named G.H. Hardy. Ramanujan fell fatally ill, and Hardy would visit him quite often.

“Hardy used to visit him, as he lay dying in hospital at Putney. It was on one of those visits that there happened the incident of the taxicab number. Hardy had gone out to Putney by taxi, as usual his chosen method of conveyance. He went into the room where Ramanujan was lying. Hardy, always inept about introducing a conversation, said, probably without a greeting, and certainly as his first remark: ‘I thought the number of my taxicab was 1729. It seemed to me rather a dull number.’ To which Ramanujan replied: ‘No, Hardy! No, Hardy! It is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways.’”

There are a few factors that are important in this story.

The first is that numbers each have their own strange properties, and as we find them, we can create new patterns that take our technology in different directions. It may seem abstract while studying math to learn that all numbers whose individual parts add up to a 3, 6, or 9 will be divisible by 3. Knowing that makes computers (and humans, of course) able to find the divisibility of 3 much faster than dividing what could be a terribly large number!

The second is that when something is the “smallest”, the efficiency in thinking and patterns is that much better. “Smallest” answers provide a much needed base for looking for the patterns. People often wonder why we should learn formulas and algorithms for finding answers. The only answer ever is that it makes it easier – all of these patterns certainly can be found by brute force – but it’s the difference between loading a dishwasher to do your dishes, or doing them by hand yourself. Which one is faster? Which one is more efficient? But you must learn how to load the dishwasher – there is a method to putting the pots in, and also one for getting the most amount of cups in.

The last thing that comes out of this is 1729 = 1^3 + 12^3 and also 1729 = 9^3 + 10^3
Therefore 9^3+10^3=1^3+12^3

Cubes are the most useful pattern we can have, because they are the 3rd-dimension of mathematics. We live in a 3-d world, and therefore finding these patterns where cubes find each other is the best for physics, 3-d rendering in computer games and graphics, and much more!