{"id":20157,"date":"2021-03-14T11:02:26","date_gmt":"2021-03-14T05:17:26","guid":{"rendered":"https:\/\/www.revoscience.com\/en\/?p=20157"},"modified":"2021-03-14T11:04:03","modified_gmt":"2021-03-14T05:19:03","slug":"after-cracking-the-sum-of-cubes-puzzle-for-42-mathematicians-discover-a-new-solution-for-3","status":"publish","type":"post","link":"https:\/\/www.revoscience.com\/en\/after-cracking-the-sum-of-cubes-puzzle-for-42-mathematicians-discover-a-new-solution-for-3\/","title":{"rendered":"After cracking the \u201csum of cubes\u201d puzzle for 42, mathematicians discover a new solution for 3"},"content":{"rendered":"\n

Jennifer Chu |<\/em><\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

CAMBRIDGE, Mass.(MIT News Office<\/em>) —\u00a0What do you do after solving the answer to life, the universe, and everything? If you\u2019re mathematicians Drew Sutherland and Andy Booker, you go for the harder problem.<\/p>\n\n\n\n

In 2019, Booker, at the University of Bristol, and Sutherland, principal research scientist at MIT, were the first to find the answer to 42. The number has pop culture significance as the fictional answer to \u201cthe ultimate question of life, the universe, and everything,\u201d as Douglas Adams famously penned in his novel \u201cThe Hitchhiker\u2019s Guide to the Galaxy.\u201d The question that begets 42, at least in the novel, is frustratingly, hilariously unknown.<\/p>\n\n\n\n

In mathematics, entirely by coincidence, there exists a polynomial equation for which the answer, 42, had similarly eluded mathematicians for decades. The equation x3<\/sup>+y3<\/sup>+z3<\/sup>=k is known as the sum of cubes problem. While seemingly straightforward, the equation becomes exponentially difficult to solve when framed as a \u201cDiophantine equation\u201d \u2014 a problem that stipulates that, for any value of k, the values for x, y, and z must each be whole numbers.<\/p>\n\n\n\n

When the sum of cubes equation is framed in this way, for certain values of k, the integer solutions for x, y, and z can grow to enormous numbers. The number space that mathematicians must search across for these numbers is larger still, requiring intricate and massive computations.<\/p>\n\n\n\n

Over the years, mathematicians had managed through various means to solve the equation, either finding a solution or determining that a solution must not exist, for every value of k between 1 and 100 \u2014 except for 42.<\/p>\n\n\n\n

In September 2019, Booker and Sutherland, harnessing the combined power of half a million home computers around the world, for the first time found a solution to 42<\/a>. The widely reported breakthrough spurred the team to tackle an even harder, and in some ways more universal problem: finding the next solution for 3.<\/p>\n\n\n\n

Booker and Sutherland have now published the solutions for 42 and 3, along with several other numbers greater than 100, this week in the Proceedings of the National Academy of Sciences<\/em><\/a>.<\/p>\n\n\n\n

Picking up the gauntlet<\/strong><\/p>\n\n\n\n

The first two solutions for the equation x<\/em>3<\/sup>+y<\/em>3<\/sup>+z<\/em>3 <\/sup>= 3 might be obvious to any high school algebra student, where x, y, and z can be either 1, 1, and 1, or 4, 4, and -5. Finding a third solution, however, has stumped expert number theorists for decades, and in 1953 the puzzle prompted pioneering mathematician Louis Mordell to ask the question: Is it even possible to know whether other solutions for 3 exist?<\/p>\n\n\n\n

\u201cThis was sort of like Mordell throwing down the gauntlet,\u201d says Sutherland. \u201cThe interest in solving this question is not so much for the particular solution, but to better understand how hard these equations are to solve. It\u2019s a benchmark against which we can measure ourselves.\u201d<\/p>\n\n\n\n

As decades went by with no new solutions for 3, many began to believe there were none to be found. But soon after finding the answer to 42, Booker and Sutherland\u2019s method, in a surprisingly short time, turned up the next solution for 3:<\/p>\n\n\n\n

5699368212219623807203<\/sup> + (\u2212569936821113563493509)3<\/sup> + (\u2212472715493453327032)3<\/sup> = 3<\/p>\n\n\n\n

The discovery was a direct answer to Mordell\u2019s question: Yes, it is possible to find the next solution to 3, and what\u2019s more, here is that solution. And perhaps more universally, the solution, involving gigantic, 21-digit numbers that were not possible to sift out until now, suggests that there are more solutions out there, for 3, and  other values of k.<\/p>\n\n\n\n

\u201cThere had been some serious doubt in the mathematical and computational communities, because [Mordell\u2019s question] is very hard to test,\u201d Sutherland says. \u201cThe numbers get so big so fast. You\u2019re never going to find more than the first few solutions. But what I can say is, having found this one solution, I\u2019m convinced there are infinitely many more out there.\u201d<\/p>\n\n\n\n

A solution\u2019s twist<\/strong><\/p>\n\n\n\n

To find the solutions for both 42 and 3, the team started with an existing algorithm, or a twisting of the sum of cubes equation into a form they believed would be more manageable to solve:<\/p>\n\n\n\n

k<\/em> \u2212 z<\/em>3<\/sup> = x<\/em>3<\/sup> + y<\/em>3<\/sup> = (x<\/em> + y<\/em>)(x<\/em>2<\/sup> \u2212 xy<\/em> + y<\/em>2<\/sup>)<\/p>\n\n\n\n

This approach was first proposed by mathematician Roger Heath-Brown, who  conjectured that there should be infinitely many solutions for every suitable k. The team further modified the algorithm by representing x+y as a single parameter, d. They then reduced the equation by dividing both sides by d and keeping only the remainder \u2014 an operation in mathematics termed \u201cmodulo d\u201d \u2014 leaving a simplified representation of the problem.<\/p>\n\n\n\n

\u201cYou can now think of k as a cube root of z, modulo d,\u201d Sutherland explains. \u201cSo imagine working in a system of arithmetic where you only care about the remainder modulo d, and we\u2019re trying to compute a cube root of k.\u201d<\/p>\n\n\n\n

With this sleeker version of the equation, the researchers would only need to look for values of d and z that would guarantee finding the ultimate solutions to x, y, and z, for k=3. But still, the space of numbers that they would have to search through would be infinitely large.<\/p>\n\n\n\n

So, the researchers optimized the algorithm by using mathematical \u201csieving\u201d techniques to dramatically cut down the space of possible solutions for d.<\/p>\n\n\n\n

\u201cThis involves some fairly advanced number theory, using the structure of what we know about number fields to  avoid looking in places we don\u2019t need to look,\u201d Sutherland says.<\/p>\n\n\n\n

A global task<\/strong><\/p>\n\n\n\n

The team also developed ways to efficiently split the algorithm\u2019s search into hundreds of thousands of parallel processing streams. If the algorithm were run on just one computer, it would have taken hundreds of years to find a solution to k=3. By dividing the job into millions of smaller tasks, each independently run on a separate computer, the team could further speed up their search.<\/p>\n\n\n\n

In September 2019, the researchers put their plan in play through Charity Engine, a project that can be downloaded as a free app by any personal computer, and which is designed to harness any spare home computing power to collectively solve hard mathematical problems. At the time, Charity Engine\u2019s grid comprised over 400,000 computers around the world, and Booker and Sutherland were able to run their algorithm on the network as a test of Charity Engine\u2019s new software platform.<\/p>\n\n\n\n

\u201cFor each computer in the network, they are told, \u2018your job is to look for d\u2019s whose prime factor falls within this range, subject to some other conditions,\u2019\u201d Sutherland says. \u201cAnd we had to figure out how to divide the job up into roughly 4 million tasks that would each take about  three hours for a computer to complete.\u201d<\/p>\n\n\n\n

Very quickly, the global grid returned the very first solution to k=42, and just two weeks later, the researchers confirmed they had found the third solution for k=3 \u2014 a milestone that they marked, in part, by printing the equation on t-shirts.<\/p>\n\n\n\n

The fact that a third solution to k=3 exists suggests that Heath-Brown\u2019s original conjecture was right and that there are infinitely more solutions beyond this newest one. Heath-Brown also predicts the space between solutions will grow exponentially, along with their searches. For instance, rather than the third solution\u2019s 21-digit values, the fourth solution for x, y, and z will likely involve numbers with a mind-boggling 28 digits.<\/p>\n\n\n\n

\u201cThe amount of work you have to do for each new solution grows by a factor of more than 10 million, so the next solution for 3 will need 10 million times 400,000 computers to find, and there\u2019s no guarantee that\u2019s even enough,\u201d Sutherland says. \u201cI don\u2019t know if we\u2019ll ever know the fourth solution. But I do believe it\u2019s out there.\u201d<\/p>\n\n\n\n

This research was supported, in part, by the Simons Foundation.<\/p>\n","protected":false},"excerpt":{"rendered":"

Jennifer Chu | CAMBRIDGE, Mass.(MIT News Office) —\u00a0What do you do after solving the answer to life, the universe, and everything? If you\u2019re mathematicians Drew Sutherland and Andy Booker, you go for the harder problem. In 2019, Booker, at the University of Bristol, and Sutherland, principal research scientist at MIT, were the first to find […]<\/p>\n","protected":false},"author":2,"featured_media":20158,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[123,17],"tags":[],"class_list":["post-20157","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematics","category-research"],"featured_image_urls":{"full":["https:\/\/www.revoscience.com\/en\/wp-content\/uploads\/2021\/03\/MIT-Cube-of-Sums-01-PRESS.jpg",900,600,false],"thumbnail":["https:\/\/www.revoscience.com\/en\/wp-content\/uploads\/2021\/03\/MIT-Cube-of-Sums-01-PRESS-200x200.jpg",200,200,true],"medium":["https:\/\/www.revoscience.com\/en\/wp-content\/uploads\/2021\/03\/MIT-Cube-of-Sums-01-PRESS-600x400.jpg",600,400,true],"medium_large":["https:\/\/www.revoscience.com\/en\/wp-content\/uploads\/2021\/03\/MIT-Cube-of-Sums-01-PRESS-768x512.jpg",750,500,true],"large":["https:\/\/www.revoscience.com\/en\/wp-content\/uploads\/2021\/03\/MIT-Cube-of-Sums-01-PRESS-675x450.jpg",675,450,true],"1536x1536":["https:\/\/www.revoscience.com\/en\/wp-content\/uploads\/2021\/03\/MIT-Cube-of-Sums-01-PRESS.jpg",900,600,false],"2048x2048":["https:\/\/www.revoscience.com\/en\/wp-content\/uploads\/2021\/03\/MIT-Cube-of-Sums-01-PRESS.jpg",900,600,false],"ultp_layout_landscape_large":["https:\/\/www.revoscience.com\/en\/wp-content\/uploads\/2021\/03\/MIT-Cube-of-Sums-01-PRESS.jpg",900,600,false],"ultp_layout_landscape":["https:\/\/www.revoscience.com\/en\/wp-content\/uploads\/2021\/03\/MIT-Cube-of-Sums-01-PRESS.jpg",855,570,false],"ultp_layout_portrait":["https:\/\/www.revoscience.com\/en\/wp-content\/uploads\/2021\/03\/MIT-Cube-of-Sums-01-PRESS.jpg",600,400,false],"ultp_layout_square":["https:\/\/www.revoscience.com\/en\/wp-content\/uploads\/2021\/03\/MIT-Cube-of-Sums-01-PRESS.jpg",600,400,false],"newspaper-x-single-post":["https:\/\/www.revoscience.com\/en\/wp-content\/uploads\/2021\/03\/MIT-Cube-of-Sums-01-PRESS-760x490.jpg",760,490,true],"newspaper-x-recent-post-big":["https:\/\/www.revoscience.com\/en\/wp-content\/uploads\/2021\/03\/MIT-Cube-of-Sums-01-PRESS-550x360.jpg",550,360,true],"newspaper-x-recent-post-list-image":["https:\/\/www.revoscience.com\/en\/wp-content\/uploads\/2021\/03\/MIT-Cube-of-Sums-01-PRESS-95x65.jpg",95,65,true],"web-stories-poster-portrait":["https:\/\/www.revoscience.com\/en\/wp-content\/uploads\/2021\/03\/MIT-Cube-of-Sums-01-PRESS.jpg",640,427,false],"web-stories-publisher-logo":["https:\/\/www.revoscience.com\/en\/wp-content\/uploads\/2021\/03\/MIT-Cube-of-Sums-01-PRESS.jpg",96,64,false],"web-stories-thumbnail":["https:\/\/www.revoscience.com\/en\/wp-content\/uploads\/2021\/03\/MIT-Cube-of-Sums-01-PRESS.jpg",150,100,false]},"author_info":{"info":["RevoScience"]},"category_info":"Mathematics<\/a> Research<\/a>","tag_info":"Research","comment_count":"0","_links":{"self":[{"href":"https:\/\/www.revoscience.com\/en\/wp-json\/wp\/v2\/posts\/20157","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.revoscience.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.revoscience.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.revoscience.com\/en\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.revoscience.com\/en\/wp-json\/wp\/v2\/comments?post=20157"}],"version-history":[{"count":0,"href":"https:\/\/www.revoscience.com\/en\/wp-json\/wp\/v2\/posts\/20157\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.revoscience.com\/en\/wp-json\/wp\/v2\/media\/20158"}],"wp:attachment":[{"href":"https:\/\/www.revoscience.com\/en\/wp-json\/wp\/v2\/media?parent=20157"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.revoscience.com\/en\/wp-json\/wp\/v2\/categories?post=20157"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.revoscience.com\/en\/wp-json\/wp\/v2\/tags?post=20157"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}