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from the solving-the-puzzle-did-not-cause-any-harm-to-mice dept.
The 21-digit solution to the decades-old problem suggests many more solutions exist.
What do you do after solving the answer to life, the universe, and everything? If you're 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 the answer to 42. The number has pop culture significance as the fictional answer to "the ultimate question of life, the universe, and everything," as Douglas Adams famously penned in his novel "The Hitchhiker's Guide to the Galaxy." The question that begets 42[*], at least in the novel, is frustratingly, hilariously unknown.
In mathematics, entirely by coincidence, there exists a polynomial equation for which the answer, 42, had similarly eluded mathematicians for decades. The equation x3+y3+z3=k is known as the sum of cubes problem. While seemingly straightforward, the equation becomes exponentially difficult to solve when framed as a "Diophantine equation" — a problem that stipulates that, for any value of k, the values for x, y, and z must each be integers.
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.
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 — except for 42.
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. 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.
Booker and Sutherland have now published the solutions for 42 and 3, along with several other numbers greater than 100, recently in the Proceedings of the National Academy of Sciences.
[*] 42: Wikipedia Entry.
Journal Reference:
Andrew R. Booker, Andrew V. Sutherland. On a question of Mordell [open], Proceedings of the National Academy of Sciences (DOI: 10.1073/pnas.2022377118)
Previously:
Sum-of-Three-Cubes Problem Solved for "Stubborn" Number 33.
(Score: 2) by bzipitidoo on Wednesday March 24 2021, @12:42PM (1 child)
Verification is often trivial compared to discovery. That's the essence of the debate about whether P=NP. If P≠NP, as most people strongly suspect, then there are all kinds of problems in which checking whether an answer is correct is easy, while finding a correct answer is hard.
A proof of the four color theorem that is an exhaustive check of all the possibilities is a perfectly valid proof whether a computer or a human did the checking. That's really what proofs are: checked all the possibilities. Found a way to cover all the cases. It's nice when there's an elegant way to break a problem down into a handful of cases, but this is not always possible. Like with this sum of cubes problem, we know that any sum s in which s%9 is 4 or 5, does not have a solution. So, no solution for 13 or 14, or 22, 23, 31, 32, etc. Why? Because any cube c%9 is always -1 (or 8), 0, or 1. There's no combination of -1,0,1 for which 3 of them can add up to 4 or -4, can only get to 3 or -3. So that's a nice, elegant elimination of 2/9ths of all the numbers, leaving the other 7/9ths an open question whether there are always 3 cubes that can sum up to them.
You could reason that, yes, a cube and the adjacent numbers always have such solutions, as they are trivial to construct c^3 = c^3 + 0^3 + 0^3 = c^3 +1^3 - 1^3. Doesn't settle a significant fraction of the possibilities, but if that's all there is, that's what you use. Potentially, thousands of such cases could combine to be a proof, and such efforts really are best done with computer assistance.
Brute force has its place. Don't discount it. Number Theory in particular is brim full of questions that are easy to ask and very hard to prove. Are there infinitely many twin primes? From the universe of Diophantine questions, Is there a perfect box? Brute force with computers can be employed to explore a larger space than we could ever hope to do by hand, and while if there are no perfect boxes it can't prove that, it could prove that there are perfect boxes by finding one.
(Score: 0) by Anonymous Coward on Wednesday March 24 2021, @05:35PM
> Verification is often trivial compared to discovery.
That's why professors grade the answers, they don't supply the answers. Their job is to tell you your work isn't good enough (unless it is then they get credit).