Sum-of-Three-Cubes Problem Solved for ‘Stubborn’ Number 33

By John Pavlus

Mathematicians long wondered whether it’s possible to express the number 33 as the sum of three cubes — that is, whether the equation 33 = x³+ y³+ z³ has a solution. They knew that 29 could be written as 3³ + 1³ + 1³, for instance, whereas 32 is not expressible as the sum of three integers each raised to the third power. But the case of 33 went unsolved for 64 years.

Now, Andrew Booker, a mathematician at the University of Bristol, has finally cracked it: He discovered that (8,866,128,975,287,528)³ + (–8,778,405,442,862,239)³ + (–2,736,111,468,807,040)³ = 33.

Booker found this odd trio of 16-digit integers by devising a new search algorithm to sift them out of quadrillions of possibilities. The algorithm ran on a university supercomputer for three weeks straight. (He says he thought it would take six months, but a solution “popped out before I expected it.”) When the news of his solution hit the internet earlier this month, fellow number theorists and math enthusiasts were feverish with excitement. According to a Numberphile video about the discovery, Booker himself literally jumped for joy in his office when he found out.

Why such elation? Part of it is the sheer difficulty of finding such a solution. Since 1955, mathematicians have used the most powerful computers they can get their hands on to search the number line for trios of integers that satisfy the “sum of three cubes” equation k = x³ + y³ + z³, where k is a whole number. Sometimes solutions are easy, as with k = 29; other times, a solution is known not to exist, as with all whole numbers that leave behind a remainder of 4 or 5 when divided by 9, such as the number 32.

But usually, solutions are “nontrivial.” In these cases, the trio of cubed integers — like (114,844,365)³ + (110,902,301)³ + (–142,254,840)³, which equals 26 — looks more like a lottery ticket than anything with predictable structure. For now, the only way for number theorists to discover such solutions is to play the mathematical “lottery” over and over, using the brute force of computer-assisted search to try different combinations of cubed integers, and hope for a “win.”

But even with increasingly powerful computers and more efficient algorithms thrown at the problem, some whole numbers have stubbornly refused to yield any winning tickets. And 33 was an especially stubborn case: Until Booker found his solution, it was one of only two integers left below 100 (excluding the ones for which solutions definitely don’t exist) that still couldn’t be expressed as a sum of three cubes. With 33 out of the way, the only one left is 42.

The reason it took so long to find a solution for 33 is that searching far enough up the number line — all the way to 1016, or ten quadrillion, and just as far down into the negative integers — for the right numerical trio was computationally impractical until Booker devised his algorithm. “He has not just run this thing on a bigger computer compared to the computers 10 years ago — he has found a genuinely more efficient way of locating the solutions,” said Tim Browning, a number theorist at the Institute of Science and Technology Austria.

Previous algorithms “didn’t know what they were looking for,” Booker explained; they could efficiently search a given range of integers for solutions to k = x³ + y³ + z³ for any whole number k, but they weren’t able to target a specific one, like k = 33. Booker’s algorithm could, and thus it works “maybe 20 times faster, in practical terms,” he said, than algorithms that take an untargeted approach.