When you look at the headline, you think:

I know the answer to this question: x, y and z are all equal to 1.

If you do a few more steps, you can see that 4, 4, and -5 are also integer solutions.Notice, these are just the first two integer solutions to the equation x cubed +y cubed +z cubed =3. What are the third integer solutions? Do you know?

In 1953, mathematician Louis Mordell asked: Does this third set of integer solutions exist?

Recently, this set of solutions was finally found.

A word of warning: never try to program using brute force!

Because these three numbers are far beyond the range of long integers, mathematicians used 400,000 computers to find the answer.

In addition, the two mathematicians have made the code open source.Of course, they are not violent searches. That’s where math comes in: it can provide you with algorithms that tell you what to search for, drastically narrowing the search space.

The discovery of whether a positive integer can be represented as the sum of three integers cubed (x³+y³+z³=k) has caused quite a stir.

This seemingly untechnical problem has puzzled the mathematical community for a long time.

Sum of three cubed numbers

In 1992, mathematician Roger Heath-Brown proposed a conjecture: for a positive integer k, if the remainder of its division by 9 is not 4 or 5 (k is not equal to 9n±4), then k can be represented as the sum of the cubes of three integers.

And every k has an infinite number of integer solutions.

For k less than 100, only k=33 and 42 have not found integer solutions before 2019.

March 2019, 33:

33 = 8866128975287528³ + (-8778405442862239)³ + (-2736111468807040)³

In September 2019, Andrew Sutherland of MIT and Andrew Booker of the University of Bristol found the answer to 42:

42 = (-80538738812075974)³ + 80435758145817515³ + 12602123297335631³

At the time, Professor Timothy Gowers of Cambridge University, who won the Fields Medal, tweeted “Congratulations”.Although the numbers below 100 were solved, there was no new solution for k=3 for decades, and many people came to believe that the so-called new solution did not exist and that the Heath-Brown conjecture was wrong.

However, after finding 42, the two Andrews soon found a third set of integer solutions for k=3, unexpectedly:

3 = 569936821221962380720³ + (-569936821113563493509)³ + (-472715493453327032)³

Mathematical Jane

To find a solution for 42 and 3, the two mathematicians started with an existing algorithm and transformed the cube and formula into a form they thought would be easier to solve:They took x+y as a parameter D, modified the algorithm further, and then divided both sides by d to find the remainder (called mod D in mathematics).

So the remainder of k divided by d is z cubed.In this way, you can be sure to find x, y, and z corresponding to k=3 by simply looking for d and z.

Even so, the digital space for searching is infinite. So they dramatically reduced the d range, bringing the search for XYZ to within 10 to the 15th power, by using the sieves of number theory.

Disassembly tasks

The two Andrewes also developed a way to split search algorithms into hundreds of thousands of parallel processing streams.

If the algorithm were run on just one computer, it would take hundreds of years to find the answer. By breaking the work into hundreds of thousands of smaller tasks, it can be run on a PC, speeding up the search even further.

In September 2019, the researchers implemented the project through Charity Engine, using the resources of ordinary users’ home computers to solve problems together.At that time, Charity Engine had more than 400,000 distributed computing computers worldwide. The Andrys deployed their algorithm on the platform.

(Note: Charity Engine also helped scientists solve a protein folding problem in an article about Science.)

Eventually, the work was divided into about 400,000 tasks, each of which took a computer about three hours to complete.

Soon, computers around the world returned the first integer solution of k=42.

And just two weeks later, they found a third integer solution for k=3, and they printed it on t-shirts.At this point, Mordell’s question 68 years ago has finally been answered.

So the question is what is the fourth solution to x cubed plus y cubed plus z cubed equal to 3?

It may not happen in your lifetime, because finding the next set of solutions would require 10 million times as many computations, four trillion computers, and it might not be enough.Andrew Sutherland, one of the authors of the paper, said, I don t know if we will ever know the fourth solution, but I am sure it exists.

[1] phys.org/news/2021-0… [2] www.pnas.org/content/118… [3] github.com/AndrewVSuth…