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Fiendishly Simple Math Problem Gets a New Solution After Puzzling World For Centuries

PETER DOCKRILL
3 APR 2019

A mathematician in the UK has found the penultimate solution to a fiendish number problem that's puzzled the world for centuries, if not even longer.

Researcher Andrew Booker, who studies pure mathematics at the University of Bristol, discovered a new solution to a famous Diophantine equation involving the sums of three cubes.

 

The problem, which has almost certainly been puzzled over for centuries by mathematicians, may date back even further, to the time of the Hellenistic mathematician Diophantus of Alexandria.

Diophantus lived during the 3rd century CE, and is credited as a kind of 'father of algebra' for his early work on algebraic and polynomial equations – although related forms of number theory date back much longer back than that, to the ancient Babylonian mathematics of millennia ago.

But enough history, what is the problem? Well, it looks like this:

x3 + y3 + z3 = k

Okay, so what we've got there is three different numbers, each being cubed, and when you add them all up, they equal a different number, which we'll call k.

There are lots of different solutions for this problem, depending on what k is, but they're not always easy to calculate. Far from it.

That's because while some (often small) values for k are easy to solve, other, sometimes bigger numerals often aren't.

This excellent video from Numberphile featuring mathematician Tim Browning, now at the Institute of Science and Technology Austria, is a fun introduction to the numerous dilemmas posed:

Because the problem can get so difficult, mathematicians started to use computers to solve the puzzle back in the 1950s, due to their ability to crunch numbers in a way that humans simply can't match.

But the problem is so fiendish, even computers have a hard time of it.

 

How hard? Well, a few years ago, a French mathematician solved the equation for k = 74, but it took a bank of computers the equivalent of slightly above 105 hours in CPU time.

Not exactly Sudoku on a Sunday afternoon.

The k = 74 breakthrough helped narrow the field down, but up until last month, when Booker released the proof of his new solution, there remained two numbers under 100 that k had never been solved for.

One of those numbers was 33. Basically, adding the cubes of x, y, and z together and getting 33 as your answer is very, very hard, it turns out, but Booker found a way to solve the problem.

This is what the first known solution for k = 33 looks like:

33 = 8,866,128,975,287,5283 +(−8,778,405,442,862,239)3 +(−2,736,111,468,807,040)3

If you want to figure out how Booker did it, take a look at his paper here, but you can also watch another Numberphile video, in which he discusses the process.

Funnily enough, it was the popularity of the original Numberphile video on the topic with Tim Browning (embedded at the top of this story) that drew both Booker and French mathematician Sander G. Huisman (who solved k = 74) to the problem in the first place.

It's a sweet example of how science communication gives back.

But none of us can rest our laurels: k = 42 has never been solved, and it's the last outstanding number under 100.

On your marks, here we go.

The findings are available on the pre-print website arXiv.org.