Unexpectedly Intriguing!
05 April 2019

Mathematicians are a strange lot.

Imagine any other profession that starts with a seemingly simple mathematical premise, but when you really get into the problem, it turns out to be so complicated that it takes decades to solve.

For example, suppose you started with a whole number as your answer, which we'll call "k". Could you work backward and identify three different integers (let's call them "x", "y", and "z") that, when cubed and summed together, would be equal to your whole number target? Here's that math in equation form:

k = x³ + y³ + z³

This kind of problem is known as a diophantine equation, but don't let the fancy name scare you - all that means is that only integers are involved in the solution.

Let's do an easy example. Let's say you wanted to find values of x, y, and z that would result in k = 36. By trial and error, you could find that setting x = 1, y = 2, and z = 3 will produce this desired result.

Seems pretty easy, right? In practice, it has proven to be anything but. In fact, since this particular challenge was first posed in 1955, mathematicians, numeric analysts, and computer programmers looking to find x's, y's, and z's to solve this equation for two-digit whole number answers have been unable to determine whether they could ever find integer values that they could cube and sum to equal either 33 or 42.

They have been able to find results for every other whole number between 0 and 100, and they've also proven that some whole numbers can never be the solution to the sum of three cubed integers, such as those that when divided by 9 leave a remainder of 4 or 5, like the whole numbers 31 or 32, which while disappointing, at least represents a confirmed outcome.

But no confirmed outcome of any kind had been found for either 33 or 42. Numberphile even presented a short video lamenting the lack of an identified solution for the lesser of these two values, entitled "The Uncracked Problem with 33":

University of Bristol mathematician Andrew Booker saw that video and decided to take on the challenge. What's more, he has successfully cracked the problem with 33. Here's his solution:

k = (8,866,128,975,287,528)³ + (–8,778,405,442,862,239)³ + (–2,736,111,468,807,040)³

And then, suddenly, only the number 42 remains of all the whole numbers less than 100 seeking a sum of three cubed integers to produce it as an answer. Numberphile has produced a new video, declaring that "42 is the new 33":

In finding these values, Booker had an edge over other mathematicians and programmers who had previously tackled the problem through the specialized computer code he developed to crunch through potential solutions:

Previous algorithms “didn’t know what they were looking for,” Booker explained; they could efficiently search a given range of integers for solutions to k = x³ + y³ + z³ for any whole number k, but they weren't able to target a specific one, like k = 33. Booker’s algorithm could, and thus it works “maybe 20 times faster, in practical terms,” he said, than algorithms that take an untargeted approach.

Diophantus of Alexandria would be proud... and so would Douglas Adams!

Bonus reading: How Search Algorithms Are Changing the Course of Mathematics, which provides a good discussion of why mathematicians are attracted to these kinds of problems!

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