After Solving the “Sum of Cubes” Puzzle for 42, Mathematicians Solve a Harder Problem That Has Stumped Experts for Years

What do you do after solving the mystery to life, the universe, and also everything? If you are mathematicians Drew Sutherland and Andy Booker, you go for the more challenging issue.

The first to discover the answer to 42 were Booker at the University of Bristol, and Sutherland, the leading research scientist at MIT, in 2019. The number has pop culture relevance as the fictional response to “the utmost question of life, the universe, and everything,” as Douglas Adams famously penciled in his unique “The Hitchhiker’s Guide to the Galaxy.” The inquiry that begets 42, at least in the book, is frustratingly, hilariously unidentified.

Coincidentally, in maths, a polynomial equation exists for which the response, 42, had likewise missed mathematicians for decades. The formula x3+ y3+ z3= k is called the sum of cubes problem. While straightforward, the equation comes to be exponentially challenging to fix when framed as a “Diophantine formula”– an issue that stipulates that, for any value of k, the worths for x, y, and z have each to be integers.

When the sum of cubes formula is framed in this way, for specific values of k, the integer solutions for x, y, and z can grow to huge numbers. The number space that mathematicians must look throughout for these numbers is larger still, needing elaborate and massive calculations.

With various means, mathematicians solved the equation over the years, either discovering a solution or establishing that a service should not exist, for each value of k between 1 and 100, excluding 42.

In September 2019, Booker and Sutherland, utilizing the power of half a million home computers combined worldwide, for the very first time found an answer to 42. The widely reported discovery stimulated the group to take on a more challenging and somewhat more universal problem: finding the following solution for 3.

The solutions for 42 and 3 and numerous other numbers higher than 100 were published by Booker and Sutherland in the Proceedings of the National Academy of Sciences.

Picking up the gauntlet

The initial two solutions for the equation x^3+ y^3+ z^3 = 3 might be evident to any high school level algebra student, where x, y, and z can be either 4, 4, and -5, or 1, 1, and 1. For years, discovering the third solution stumped professional number theorists. In 1953, the problem triggered pioneering mathematician Louis Mordell to question: Is it even possible to verify whether other solutions for 3 exist?

According to Sutherland, this was kind of like Mordell throwing down the gauntlet. The interest in solving this problem is not as much for the particular solution as it is to better understand just how challenging these equations are to solve. It is a benchmark against which we can measure ourselves.”

As years passed with no new solutions for 3, many started to think there were none to be discovered. However, not long after finding the solution to 42, Booker and also Sutherland’s technique, in a short time, showed up the following solution for 3:

569936821221962380720^3 + (− 569936821113563493509)^3 + (− 472715493453327032)^3 = 3

The discovery was a straightforward solution to Mordell’s problem: Yes, it is feasible to discover the next solution to 3, and also, what is more, right here is that solution. Furthermore, perhaps more generally, the solution, involving enormous, 21-digit numbers that were not possible to sort out until now, suggests that there are many more solutions around, for 3, and other values of k.

“There had been some severe uncertainty in the mathematical and computational communities because [Mordell’s question] is tough to test,” Sutherland says. “The numbers quickly get quite big. You never will find more than the first solutions. However, what I can state is, upon finding this solution, I am convinced there are infinitely more out there.”

A solution’s twist

To find the solutions for 42 and 3, the group began with an existing formula, or a twist of the sum of cubes equation right into a way they thought would be simpler to address:

k − z^3 = x^3 + y^3 = (x + y)( x^2 − xy + y^2)

This approach was first suggested by mathematician Roger Heath-Brown, who judged that there should be infinite solutions for each suitable k. The team further altered the formula by representing x+ y as the parameter d. They then reduced the equation by splitting both sides by d and keeping just the remainder – a mathematical operation labeled “modulo d”- leaving a simplified representation of the issue.

While explaining, Sutherland asks to consider k as a cube root of z, modulo d. This way, think of working in an arithmetic system where you worry about the remainder modulo d, and they are trying to calculate a cube origin of k.

With this sleeker variation of the equation, the researchers would only be required to search for d and z, which would undoubtedly ensure locating the supreme solutions to x, y, and z, for k= 3. However, the space of numbers they would certainly need to explore would undoubtedly be considerably huge.

So, the scientists improved the algorithm by using mathematical “sieving” strategies to reduce the space of possible solutions for d substantially.

“This includes some relatively sophisticated number theory, making use of the framework of what we understand about number fields to prevent searching in areas we do not require to look,” Sutherland claims.

A worldwide task

The group likewise developed methods to efficiently divide the algorithm’s search into hundreds of thousands of parallel processing streams. If the algorithm were executed on just one computer, it would have taken hundreds of years to discover a solution to k= 3. By separating the work into countless smaller tasks, each separately executed on a separate computer, the group could also speed up their search.

In September 2019, the scientists put their strategy into play with Charity Engine, a project that can be downloaded and installed as a cost-free app by any personal computer. The app is designed to harness any spare home computing power to solve complex mathematical problems collectively. At the time, Charity Engine’s grid made up over 400,000 computer systems around the world, and Booker and Sutherland could execute their algorithm on the network as a test of Charity Engine’s brand-new software program system.

“For each computer system in the network, they are told, ‘your work is to seek d’s whose prime factor drops within this range, subject to a few other conditions,’ Sutherland claims. “Furthermore, we needed to discover how to separate the work upright into roughly 4 million three-hour tasks for a computer to complete.”.

Extremely swiftly, the global grid returned the first solution to k= 42, and simply two weeks later on, the scientists verified they had found the third solution for k= 3 – a landmark that they marked, partially, by printing the equation on t-shirts.

The third solution to k= 3 existence suggests that Heath-Brown’s original opinion was correct, which there are more services yet the most recent one. Heath-Brown likewise predicts the area between solutions will undoubtedly grow exponentially, together with their searches. For example, rather than the third solution’s 21-digit values, the fourth solution for x, y, and z will likely involve numbers with an astounding 28 digits.

Sutherland continues by saying that the amount of work you have to do for each new solution expands by a factor of greater than 10 million, so the following solution for three will require 10 million times 400,000 computers to discover and no warranty even enough. He finalizes, stating that he does not know if they will ever know the fourth solution, but he believes it is around.

Originally published on Sciencedaily.com. Read the original article.

Reference: “On a question of Mordell” by Andrew R. Booker and Andrew V. Sutherland, 10 March 2021, Proceedings of the National Academy of Sciences.
DOI: 10.1073/pnas.2022377118