Too often, students view math as a stationary field, one that was invented (or discovered, depending on your perspective) hundreds of years ago and has remained unchanged since, with everything knowable having been figured out long ago. It’s easy to think that the only questions that remain are the incredibly difficult, abstract, esoteric ones that would require decades of math study to even understand. But that’s simply not true!
Consider the equation x³ + y³ + z³ = k. Easily understood: take three integers {x, y, z}, cube them, and add them together. In 1955, mathematicians at the University of Cambridge asked if a set of {x, y, z} could be found to add to every positive integer k less than 100. Some were easy to find: (-5)³ + 7³ + (-6)³ = 2; 2³ + (-3)³ + 4³ = 45; 25³ + (-17)³ + (-22)³ = 64. But others proved surprisingly challenging, requiring cubes of much larger numbers in order to form (51 is the sum of the cubes of -796, 659, and 602, and the solution for 30, found only in 1999, required the cubes of 2,220,422,932, -2,218,888,517, and -283,059,965). Even more unfortunately, there appeared to be not much of a pattern in the trios of numbers that worked, and so finding new solutions mostly amounted to an enormous guess-and-check procedure. A pair of mathematicians proved in 1979 that any number that the expressions 9n – 4 or 9n + 4 evaluate to (4, 5, 13, 14, 22, 23, 31, 32, 40, 41, 49, 50, 58, 59, 67, 68, 76, 77, 85, 86, 94, and 95) could not be expressed as the sum of three cubes, which took out several elusive numbers, but the search wore on to finish the list. Until this year.
At the start of 2019, a sum of cubes had been found every possible positive integer k < 100 had been found except for two: 33 and 42. Computer programmers had been checking possible combinations for more than 50 years, and had progressed with their checks up to numbers beyond 100 trillion, but with no success! Finally, Andrew Booker, a mathematician at the University of Bristol, came up with a new way to search for likely trios much more efficiently and set a university supercomputer to the task. It took only three weeks to find a solution:
33 = (8,866,128,975,287,528)³ + (–8,778,405,442,862,239)³ + (–2,736,111,468,807,040)³
When he found the solution, Booker said he literally jumped for joy. But his job wasn’t done! There was still one number to be solved, and he knew this task would be too large for even his university’s computer.
So he turned to MIT’s Andrew Sutherland and a worldwide computer group called Charity Engine, Members of the group from around the world run a program that uses their computers’ downtime to do data crunching, effectively donating their devices’s computing time to a variety of causes. Fans of Douglas Adam’s The Hitchiker’s Guide to the Galaxy will notice a similarity here to the story, where a computer the size of a planet is constructed to find the “Ultimate Question of Life, the Universe, and Everything,” which a previous supercomputer had identified the answer as 42.
After a combined computing time equivalent to almost 150 years, they found the answer this month:
42 = (-80,538,738,812,075,974)³ + (80,435,758,145,817,515)³ + (12,602,123,297,335,631)³
The next-lowest number to have an unknown sum of cubes is 114, and in fact there are only ten numbers less than 1000 for which such a solution is unknown.
If you’re interested in learning more about this mathematical puzzle, I’d suggest you start with this Numberphile video that Booker says was his inspiration to start on his hunt. You can see an interview with Booker after his cracking of 33 here, and a recent followup here.
MrKirkMath.com 