Unsolvable maths problem, solved

MATHEMATICIAN Andrew Booker was searching for topics to present at his children’s maths club when he stumbled on a YouTube video about the sum-ofthree-cubes puzzle. ‘It was a case of serendipity,’ the University of Bristol mathematics professor says. The video – produced by the popular YouTube channel Numberphile – led him to solve a curious equation that has stumped mathematicians for decades.

In 1825, mathematician S Ryley proved in the magazine Ladies’ Diary that any fraction can be represented as the sum of three cubes of fractions. Mathematician Louis Mordell took the puzzle a step further in 1953, when he questioned whether the same type of solution could be found for x3 + y3 + z3 = k, a Diophantine equation, which involves only positive and negative integers. He started with the number 3, and the hunt was on to find solutions for all integers between 1 and 100.

In some cases, finding the solution to x3 + y3 + z3 = k is easy enough.

To reach 53, for example, all that’s needed is 33 + 33 + (–1)3. Other numbers, such as 51, grow increasingly complicated: (–796)3 + 6593 + 6023. For others – 4, 5, 13, 14, 22, 23, 31, 32 – it is mathematically impossible. Then there are the numbers that verged on unsolvable but seemed hypothetically possible, such as 33 and 42.

In March 2019, Booker cracked 33. The computer he used searched through 10 quadrillion numbers and turned up three within a matter of weeks: 8 866 128 975 287 5283, (–8 778 405 442 862 239)3, and

(–2 736 111 468 807 040)3. Encouraged, he reached out to Andrew Sutherland of MIT to see if he might help solve for 42. The number has a mythical reputation in pop culture – most notably in Douglas Adams’s Hitchhiker’s Guide to the Galaxy, but also within the works of Lewis Carroll, author of Alice in Wonderland (and a mathematician). Sutherland agreed. The chance to find a set of mystery numbers equaling 42 was irresistible, but they needed more juice.

Then, Mark McAndrew, founder of Charity Engine, a supercomputing system that harnesses unused computing power from more than 500 000 home PCs, caught a Numberphile video about Booker’s quest for

42. He wanted in, too. ‘I was literally jumping around,’ he says. Over the course of a day, the vast network of PCs logged more than a million computing hours on Charity Engine’s volunteer computer network. Finally, they got their answer: (–80 538 738 812 075 974)3

+ 80 435 758 145 817 5153

+ 12 602 123 297 335 6313.

‘The team’s discovery is a major piece of progress on this problem,’ says mathematician Chao Li of Columbia University. ‘It is always exciting and satisfying in mathematics to see a long-standing problem finally being resolved.’ (Booker and Sutherland also recently found a new way to make a sum of three cubes equal 3.) Sutherland knows that when problems are solved with the help of supercomputers, that’s all the public tends to see. ‘But the way these things work, there’s always an interplay between algorithm and computational power,’ he says. Supercomputers are like tractors, threshers, or any other machine. All the power in the world is great, but without knowing how to use it, you’re stuck – be it in a field or amid 10 quadrillion numbers.

IT IS ALWAYS EXCITING AND SATISFYING IN MATHEMATICS TO SEE A LONGSTANDING PROBLEM FINALLY BEING RESOLVED.

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A worldwide network of computers helped solve one of mathematics’ most elusive problems.