Mathematician solves 64-year-old ‘diophantine puzzle’

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

Andrew Booker, reader of pure mathematics at the University of Bristol in the UK, discovered how to express the number 33 as the sum of three cubes.

Mathematicians have been trying to solve this specific puzzle since at least 1955. And more generally, these types of problems fall into 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 back as Babylonian mathematics 4,000 years ago – and their study has emerged independently in several cultures. old ones,” Booker said. Newsweek. “Apart from historical interest, Diophantine equations have proven to be a very rich field of study for mathematicians.”

“On the one hand, they’re appealing because of their apparent simplicity — ultimately, they’re just questions about ordinary whole numbers,” he said. “On the other hand, their solutions can display mind-boggling levels of complexity – they can range from easy to impossible to solve.”

Despite enormous advances in computer technology over the past few decades, there remained two whole numbers less than 100 – 33 and 42 – that no one had been able to express as the sum of three cubes. (There is a group of numbers that are known to have no solution, those that leave a remainder of 4 or 5 when divided by 9.)

But Booker has now shown that the number 33 can be expressed as: (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 solve 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. “It made me want to try!”

And with the help of a computer, he was able to identify the solution: “I found a search algorithm that refines previous methods and allows us to target a specific number, like 33 or 42, more efficiently,” said he declared. “The short article I wrote on the subject focuses mainly on the theoretical analysis of these refinements, but a large part of it also on the ever-increasing computing power. Improvements in algorithms and the law of Moore are what allows us to reach the beaches that we can today.”

“I had a pretty good guess that I would find something for one of the numbers below 1000,” he said in the statement. “But I didn’t know it would be the number 33. We don’t know if the remaining numbers have infinite solutions, or how often those solutions are. It’s quite mysterious.”

Some mathematicians have suggested that some of the remaining unsolved numbers – aside from the group that leaves a remainder of 4 or 5 – are simply unsolvable. However, developments like the last solution suggest that all eligible numbers have a solution.

“Some numbers — the ones that leave the remainder 4 or 5 when you divide them by 9 — don’t have a solution, and that’s something we can prove,” Booker said. Newsweek. “With the exception of these, we assume that all other numbers should have solutions, but they may be very rare. This rarity is what makes solutions difficult to find and also difficult to analyze theoretically – at least point that the question might be undecidable.”

“Many numbers have solutions that aren’t too hard to find, but every once in a while you come across a stubborn one that resists all of our attempts to find it,” he said. “For example, now that 33 have been found, there are still 12 numbers below 1,000 with no known solutions.”

The mathematician is now working with Andrew Sutherland of MIT to try to find the solution for the final unsolved number less than 100—42.

This article has been updated to include additional comments from Andrew Booker.

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