Taxicab numbers: why 1729 is the number it is

0:00Here is a number. One thousand seven hundred and twenty-nine. Say nothing about it, and it looks like nothing. Four digits, no obvious shape, not a square, not a cube. A mathematician once looked at exactly this number and called it dull. He was wrong, and the way he was wrong is worth ten minutes. Because a number is never dull. Only the looking is. The story is short. Srinivasa Ramanujan was ill in hospital in Cambridge.

0:35His colleague Godfrey Hardy came to visit, arrived in a taxi, and made small talk about the number painted on it. One thousand seven hundred and twenty-nine, he said. Rather a dull number. He hoped it was not a bad sign. Ramanujan disagreed immediately, and he did not say it was a nice number, or an interesting one. He said something far more precise than that, and the precision is the whole lesson.

1:08What he said was that it is the smallest number that can be written as two cubes added together, in two different ways. Start with the two ways. One cubed is one. Twelve cubed is one thousand seven hundred and twenty-eight. Add them and you land exactly on the taxi. That is the first way. Now leave those two alone and take a different pair. Nine cubed is seven hundred and twenty-nine.

1:37Ten cubed is one thousand. Seven hundred and twenty-nine plus one thousand is one thousand seven hundred and twenty-nine again. Two completely different pairs of cubes, one destination. Before going on, one thing has to be ruled out, because it is the cheap way to make any number look special. Twelve cubed plus one cubed is not a third way. It is the first way written backwards. If you count arrangements rather than pairs you get four for this number, and four is not the claim.

2:15The claim is about pairs, and there are two. One and twelve. Nine and ten. No number appears in both. Now look again at what Ramanujan actually said, because the important word is not cubes. It is smallest. Those two sentences are not the same size of statement. Saying this number splits two ways is a PROPERTY. You can settle it about one number, on your own, in a minute.

2:46Take one cube off, ask whether what is left is a cube, and repeat. For this number there are nine cubes small enough to be worth trying, so nine questions and you are done. Saying it is the smallest that splits two ways is a RECORD. A record is not about one number at all. It is a statement about every number underneath it, all one thousand seven hundred and twenty-eight of them, and you do not own it until you have ruled out the lot.

3:19So can that be done? It can, and the reason is that the cubes run out fast. If two cubes add to something no bigger than our number, then neither cube can be bigger than it either. Twelve cubed is one thousand seven hundred and twenty-eight, which just fits. Thirteen cubed is two thousand one hundred and ninety-seven, which does not, on its own, before anything is added to it.

3:47So the only cubes in play are the twelve from one cubed up to twelve cubed. And the smaller of the pair is squeezed harder still. Two nines cubed already come to one thousand four hundred and fifty-eight, and two tens cubed overshoot. So the smaller cube never gets past nine. That turns the record into something you can finish. Do not go number by number. Go pair by pair.

4:16Take one cubed, and add each cube in turn: two, nine, twenty-eight, sixty-five, and on up to one thousand seven hundred and twenty-nine. Twelve sums in that row. Take two cubed and do the same, and there are ten sums before you leave the range. Then nine, then eight, seven, six, five, three, two. Add up those rows. The whole question is sixty-two sums. Not sixty-two thousand. Sixty-two. Now the finish is pure counting, and it needs no cleverness at all.

4:55Those sixty-two sums land on sixty-one different totals. Sixty-two arrivals, sixty-one places. So somewhere, exactly once, two of them landed on the same number. One collision. And when you look at where it happened, it happened at one thousand seven hundred and twenty-nine. That is the record, earned. Nothing below it does this, because everything below it is on that list, and every total on that list came up once.

5:28Something else falls out of the same list, and it is worth stopping on. Below the taxi there are one thousand seven hundred and twenty-eight numbers. Sixty of them are two cubes added, in any way at all. One thousand six hundred and sixty-eight are not. So being a sum of two cubes even once is already rare. The first few that manage it are two, nine, sixteen, twenty-eight, thirty-five, fifty-four.

5:57And of those sixty, exactly one manages it twice. It does not get easier further out, either. Below ten thousand, two hundred and two numbers are a sum of two cubes; below a hundred thousand, nine hundred and thirty-eight. The count grows and the share shrinks. Here is the mistake this story invites, and it is worth killing straight away. The taxi number is not the only number that does this.

6:28It is the first. The second is four thousand one hundred and four. The third is thirteen thousand eight hundred and thirty-two. And the family never stops, for a reason you can see in one line: multiply a member by any cube and both of its ways come along, scaled. Twenty-seven times the taxi is forty-six thousand six hundred and eighty-three, and it is three cubed plus thirty-six cubed, and also twenty-seven cubed plus thirty cubed.

7:02But look how thin it is at the start: the second one is more than twice the first. Those two are yours to crack. Each of them splits into two cubes in two ways. You have the method, you have the strip of cubes, and the answers are not going on this board. Now watch how much a single word in that claim is holding up. Two POSITIVE cubes. Take that word out, and let a cube be negative.

7:32Ninety-one is three cubed plus four cubed: twenty-seven and sixty-four. Ninety-one is also six cubed plus minus five cubed: two hundred and sixteen, take away one hundred and twenty-five. Two ways. And ninety-one is nowhere near the taxi. Drop that one word and eleven numbers below the taxi do it twice, and the record is simply gone. Losing the record is not even the worst of it. Losing the word also breaks the search.

8:08The neat table worked because a cube bigger than the target was out of the question. With a minus sign that reasoning dies, because the other cube can cancel most of it away. Below the taxi, the biggest cube you need is twenty-four cubed, which is thirteen thousand eight hundred and twenty-four. That is almost eight times the number you are testing. So the old walk, out to twelve and stop, finds nine of the eleven and quietly misses two.

8:40A search with the wrong boundary does not announce that it stopped early. It just comes back and tells you there is nothing there. So back to the doorway. What Ramanujan did was not fast arithmetic. Nobody establishes a record standing in a doorway. He already knew, the way you know a friend's face, and another colleague, John Littlewood, said exactly that: that every whole number seemed to be a personal acquaintance of his.

9:13And the number really was not dull. It is seven times thirteen times nineteen: three primes, each six past the last. Its digits add to nineteen, which is one of those three. So it divides by its own digit sum, and the answer is ninety-one, which is the number we met a minute ago. None of that makes it special. That is the point. Look at any number long enough and it stops being dull, and the only thing Hardy was really reporting was how long he had looked.