The Uncracked Problem With 33: Mathematician Solves 64-year-old 'Diophantine Puzzle'

An American mathematician has cracked part of a problem that had remained unsolved for 64 years.

Andrew Booker, Reader of Pure Mathematics at the University of Bristol in the U.K., worked out how to express the number 33 as the sum of three cubes.

Mathematicians have been trying to solve this specific condundrum since at least 1955. And more generally, these types of problems fall under a category known as "Diophantine equations."

"They are named after the mathematician Diophantus, who lived in the 3rd century AD, but many special cases go back much further—as far as Babylonian mathematics 4,000 years ago—and their study arose independently in several ancient cultures," Booker told Newsweek. "Besides the historical interest, Diophantine equations have proved to be a very rich area of study for mathematicians."

"On the one hand, they are seductive because of their apparent simplicity—at the end of the day, they're just questions about ordinary whole numbers," he said. "On the other, their solutions can display mind-boggling levels of complexity—they can range from easy to impossible to solve."

Despite huge advances in computer technology in recent decades, there remained two whole numbers under 100—33 and 42—that nobody had been able to express as the sum of three cubes. (There are a group of numbers which are known to have no solution—those which leave a remainder of 4 or 5 when divided by 9.)

But Booker has now shown that the number 33 can be expressed like so: (8,866,128,975,287,528)³ + (–8,778,405,442,862,239)³ + (–2,736,111,468,807,040)³.

The mathematician was inspired to try and tackle the problem after watching a YouTube video featuring a former professor at Bristol, Tim Browning.

"The video was called 'The Uncracked Problem,'" Booker said in a statement. "That made me want to have a go!"

And with the help of a computer, he was able to identify the solution: "I came up with a search algorithm that refines earlier methods and allows us to target a specific number, like 33 or 42, more efficiently," he said. "The short paper that I wrote on the topic focuses mostly on the theoretical analysis of those refinements, but a big part of it also ever-increasing computer power. Improvements in algorithms and Moore's Law together are what allow us to reach the ranges that we can today."

"I had a pretty good guess that I'd find something for one of the numbers below 1000," he said in the statement. "But I didn't know it was going to be the number 33. We don't know if the remaining numbers have infinitely many solutions, or how frequent those solutions are. It's quite mysterious."

Some mathematicians have suggested that some of the remaining unsolved numbers—aside from the group which leave a remainder of 4 or 5—are simply impossible to solve. However, developments like the latest solution suggest that all eligible numbers have a solution.

"Some numbers—those that leave remainder 4 or 5 when you divide them by 9—have no solutions, and that's something we can prove," Booker told Newsweek. "Except for those, we guess that every other number should have solutions, but that they can be very sparse. That sparseness is what makes the solutions hard to find and also hard to analyze theoretically—to the point that the question could be undecidable."

"Many numbers have solutions that are not too hard to find, but every so often you come across a stubborn one that resists all our attempts to find it," he said. "For example, now that 33 has been found, there are still 12 numbers below 1,000 with no known solutions."

The mathematician is now working with Andrew Sutherland of MIT in an attempt to find the solution for the final unsolved number below 100—42.