One million hours.
That’s how long it took to run the calculations for a ground-breaking mathematics discovery. And that wasn’t some lone genius typing numbers into his graphing calculator for a million hours: it was a worldwide computer network running calculations for a million hours. And that was after mathematicians have spent decades working on the problem. All to figure out how to calculate the number forty-two. And before that thirty-three.
What gives?
Introducing Diophantine equations
The problem in question is something called a Diophantine equation, which Encylopedia Britannica defines as “involving only sums, products, and powers in which all the constants are integers and the only solutions of interest are integers.” The most famous example is probably the Pythagorean Theorem, a2+b2=c2. But Diophantine equations are a popular topic in the mathematics field of number theory because there are so many of them, and because so many of them turn out to be trickier than expected.
“I like the fact that Diophantine equations appear so simple, but changing one small part of the equation can dramatically change how complicated the solutions are,” Lillian Pierce, the Nicholas J. and Theresa M. Leonardy Associate Professor of Mathematics at Duke University, told TIP.
“For example,” she explained, “look at the equation x2-1=0. We know that the solutions to this are the very familiar integers +1 and -1. Let’s change the equation to x2-2=0. In order to solve this, we need to have something called ‘the square root of 2.’ We know what that is nowadays, but there’s something special about that number, because it’s not a whole number, and it’s not even a rational number. So in order to solve this new equation, we have to ‘invent’ irrational numbers.
“Let’s change the equation again to x2+1=0. Now in order to solve this equation, we have to ‘invent’ imaginary numbers, so that we have a square root of -1. All the equations look roughly as “complicated” as each other, but by making these small changes, we arrived at equations where we needed a deeper understanding of what type of number we would allow as a solution.”
More generally, Diophantine equations are an example of something called number theory, which Idris Assani, a professor of mathematics at the University of North Carolina at Chapel Hill, tells TIP is the “golden field” of mathematics. “In many, many cases, the source [of the problems mathematicians are working on] is a problem in number theory.”
The summing of three cubes
What does this have to do with the number 42? That comes from another famous Diophantine equation called the “summing of three cubes.” It’s “a math puzzle has stumped the smartest mathematicians in the world,” according to Popular Mechanics.
The problem involves taking x3+y3+z3=k and finding the values for x, y, and z such that k equals all whole numbers. Some k values are easy, like 1 (13+03+03) and 3 (13+13+13). Even 29 is pretty simple (33+13+13).
Others are impossible—but mathematicians have been able to prove that they are impossible. In fact, as mathematician Tim Browning explains in the video above, they have found that all the k values that can be written as 9k+4 or 9k+5 are impossible to solve for.
But there were a few other numbers that no one could find a solution to and that they couldn’t prove were impossible to solve. In 2015, when Browning recorded the video, the smallest k value they couldn’t figure out was the number 33.
But in March of this year, University of Bristol mathematician Andrew Booker solved it. It’s 8,866,128,975,287,528³+(-8,778,405,442,862,239)³+(-2,736,111,468,807,040)³.
But Booker wasn’t content with figuring out the solution for 33. After that discovery, he set his sights on the next lowest number no one could figure out: 42. And two months ago, he cracked that one, too.
The answer? (-80,538,738,812,075,974)³+80,435,758,145,817,515³+12,602,123,297,335,631³.
Booker and the Charity Engine
Booker’s success was due to both technology and some creative thinking.
“Because of the limits of our brains, there are computations you cannot go beyond,” Assani said. That’s where computers come in. “The computer can do those computations very fast and give you some insight on how to solve the problem.”
But the answers to the summing of three cubes are so big, that normal computers can’t hope to run the calculations. Instead, Booker used something called the Charity Engine, which is a program individuals around the world can install, allowing researchers like Booker to take advantage of their unused computer power to run big, complicated complications—like looking for three 17-digit numbers that you can cube and add together to get 42—without wasting a ton of energy and money.
But brute force wasn’t enough, either. The numbers involved are so, so big that even modern supercomputers like the Charity Engine can’t keep up. Booker’s solution was to think of a more efficient algorithm he could run, which allowed the Charity Engine to search even bigger numbers. You can hear he describe his approach in the video above.
Doing so helped Booker solve a decades old equation—but he says the approach won’t work for the next numbers that don’t have a solution. (The current smallest is 114.) “I don’t think these are sufficiently interesting research goals in their own right to justify large amounts of money to arbitrarily hog a supercomputer,” Booker told Quanta Magazine.
Why bother?
Booker even admitted that his wife wasn’t sure why she should care about solving this problem. So why should you?
Part of the reason is that Diophantine equations are useful in general. “They’re rich enough to encode [other mathematical] statements that have nothing to do with Diophantine equations,” one mathematician told Quanta. “They’re rich enough to simulate computers.” That means they have practical implications for things like cryptography, according to the website Brilliant. The summing of three cubes is particularly important because it’s on the simpler end of those Diophantine equations, which means it’s at the limit of what we can currently work on.
But more generally, it’s important because it’s the way math works. Typically, the field of math is divided up into “pure” math—like proving number theory theorems—and “applied” math—like the math you do in physics or engineering. But the distinction is less of a hard line and more of a spectrum, Katie Newhall, another professor of mathematics at the University of North Carolina, tells TIP.
“You need the pure math theorists to let you know things like the rules of mathematics you’re allowed to follow,” Newhall says. “Then the person in the middle says, Great, now I can follow those rules and recombine them in new ways to bring some understanding to an experimental system. And then you need the applied math experimentalist to do an experiment and see what happens, and then see what the math might help you predict.”
Booker’s work is one small piece of the pure math side. “It’s like an arrow with a cord you throw in the dark and see if it hits the target,” Assani says. “When it does hit the target, you hear a Bam! And then you say, There was a bam, now I follow the cord.” In other words, mathematicians can take this one success and use it to start building a theory. If nothing else, the success means you have a data point to go on.
It can still seem like a very, very small victory. But that’s because the math most people are familiar with has already been distilled into useful rules. “Isaac Newton and those other mathematicians were thinking about chopping up shapes into tiny shapes and cubes that weren’t the right shape but were close to it,” Newhall says, but eventually that thinking ended up creating calculus—the kind of math people learn in high school and use to study real-world topics like velocity and acceleration. Advances like Booker’s are at an earlier stage, in which we’re still trying to piece together what those rules might be.
The more we learn about that problem, the more we might learn about the problems in general—and that may provide important future insights.
Lauren G Shaw says
Good article and explanation of why I should care. But there’s a typo in your article. 29 is not 9 cubed plus 1 plus 1. It should be 3 cubed.
Duke TIP says
Thank you for catching our silly error! We’ve corrected it.