Sum-of-Three-Cubes Problem Solved for ‘Stubborn’ Number 33:
A number theorist with programming prowess has found a solution to 33 = x³ + y³ + z³, a much-studied equation that went unsolved for 64 years.
Mathematicians long wondered whether it’s possible to express the number 33 as the sum of three cubes — that is, whether the equation 33 = x³+ y³+ z³ has a solution. They knew that 29 could be written as 3³ + 1³ + 1³, for instance, whereas 32 is not expressible as the sum of three integers each raised to the third power. But the case of 33 went unsolved for 64 years.
Now, Andrew Booker, a mathematician at the University of Bristol, has finally cracked it: He discovered that (8,866,128,975,287,528)³ + (–8,778,405,442,862,239)³ + (–2,736,111,468,807,040)³ = 33.
Booker found this odd trio of 16-digit integers by devising a new search algorithm to sift them out of quadrillions of possibilities. The algorithm ran on a university supercomputer for three weeks straight. (He says he thought it would take six months, but a solution “popped out before I expected it.”) When the news of his solution hit the internet earlier this month, fellow number theorists and math enthusiasts were feverish with excitement. According to a Numberphile video about the discovery, Booker himself literally jumped for joy in his office when he found out.
Why such elation? Part of it is the sheer difficulty of finding such a solution. Since 1955, mathematicians have used the most powerful computers they can get their hands on to search the number line for trios of integers that satisfy the “sum of three cubes” equation k = x³ + y³ + z³, where k is a whole number. Sometimes solutions are easy, as with k = 29; other times, a solution is known not to exist, as with all whole numbers that leave behind a remainder of 4 or 5 when divided by 9, such as the number 32.
[...]33 was an especially stubborn case: Until Booker found his solution, it was one of only two integers left below 100 (excluding the ones for which solutions definitely don’t exist) that still couldn’t be expressed as a sum of three cubes. With 33 out of the way, the only one left is 42.
Next up is 42? Where have I seen that number before?
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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.
(Score: 5, Funny) by FatPhil on Wednesday March 27 2019, @11:46AM (6 children)
It would be nice to see the other low-height solutions cracked, but it would be most pleasing if 42 were the last to yield.
Great minds discuss ideas; average minds discuss events; small minds discuss people; the smallest discuss themselves
(Score: 4, Funny) by driverless on Wednesday March 27 2019, @12:41PM (4 children)
Man, these mathematicians really need to get out more.
(Score: 2) by VLM on Wednesday March 27 2019, @01:22PM
In the dick size war that is academic mathematics, (8,866,128,975,287,528)³ is pr0n star large number material. I can only calculate up to 6 in my head. More caffeine would probably help. I'm not an idiot I figured out 8 pretty easily LOL but I had a mental block or whatever requiring looking up 7 online, which was kinda obvious once I saw it. I think most people could get up to 16 without a computer, 16 is a tough one.
Also there's the primate dominance achievement of bending this monster to your will, regardless how useless the result:
https://www.bristol.ac.uk/acrc/high-performance-computing/bluecrystal-tech-specs/ [bristol.ac.uk]
For some homelab p0rn, here's some images from my basement:
https://www.acrc.bris.ac.uk/gallery/gallery1.htm [bris.ac.uk]
(no actually I'm kidding, thats not my basement, I do have a half rack under my basement stairs harry-potter-bedroom style. I've worked in many facilities vaguely reminiscent of that HPC center. Nice halon plumbing in the background of the pix, that must be like $100K of halon alone.)
(Score: 3, Funny) by slinches on Wednesday March 27 2019, @04:48PM (2 children)
Not sure what took them so long to solve this problem, either. There are an effectively infinite number of solutions and it took me under a minute to come up with this answer:
1^3+2^3+2.8845^3=33.000
That's close enough, at least. Anyone who needs more than five significant digits of precision is measuring the wrong thing.
-- An Engineer
(Score: 2) by opinionated_science on Wednesday March 27 2019, @06:30PM (1 child)
I modded you as funny, as we all know only decimals ".0" count ;-)
(Score: 0) by Anonymous Coward on Thursday March 28 2019, @09:19AM
Easy:
x=y=z=11^(1/3)
See? No decimal places anywhere. ;-)
(Score: 2) by bob_super on Wednesday March 27 2019, @06:56PM
Easy :
42 = angels3 + head3 + pin3
(Score: 2) by looorg on Wednesday March 27 2019, @02:39PM
Great news everybody! ... and then all the problems started to appear. It's just A solution, it's not THE solution. There could be many, or even an infinite amount of solutions. Also it's just a solution, it's not like it's a proof or theorem. Perhaps one day one can be constructed with knowledge gained from this solution but I wouldn't hold my breath. Which is usually the issue or problem with a lot of these computer generated solutions, they just add very little to the knowledge base. At best it grants us a little extra insight, that perhaps one day in the future will allow us to understand something more but that is quite hard to say. Booker even acknowledge that, but the writer somehow decided to bury that info about 2/3 into the article at the end of a paragraph.
(Score: 5, Funny) by DannyB on Wednesday March 27 2019, @02:54PM (2 children)
I recently watched a Numberphile video [youtube.com] showing a math teacher attempting to find a way to cut a ham sandwich in half. It took almost six minutes to present a way to do it.
I don't think it's that hard of a problem.
New problem . . .
Teacher: I am going to teach you how to evenly cut a round cake so that it may be eaten by any number of people. The two requirements are:
1. The cut cake pieces must be extremely close to equal in size.
2. There must be nobody who complains that their piece is too small.
First, you take 360 degrees and divide by the number . . . .
Student: (interrupting) Your approach is too complicated. If three people were wanting to eat the cake, a much simpler solution would be to cut the cake in half (as closely as possible) and shoot the third person. That way both conditions (1) and (2) are satisfied.
Teacher: Ah, but your approach required an additional tool, the gun.
Student: My method can be simplified by using only one tool, the knife. The third person is stabbed instead of shot. Whether this step is performed before or after cutting the cake is a matter of preference.
Teacher: I see your approach is simpler and worthy of a Fields medal.
If you think a fertilized egg is a child but an immigrant child is not, please don't pretend your concerns are religious
(Score: 2) by Osamabobama on Wednesday March 27 2019, @10:16PM (1 child)
I wonder how many people throughout history have stabbed somebody in order to win a Fields medal...
Appended to the end of comments you post. Max: 120 chars.
(Score: 2) by DannyB on Thursday March 28 2019, @03:06PM
I don't believe the fields medal existed at the time of the guy who shot Galois. But then that was not a stabbing.
If you think a fertilized egg is a child but an immigrant child is not, please don't pretend your concerns are religious
(Score: 0) by Anonymous Coward on Wednesday March 27 2019, @03:56PM
Ah yes, taking a (cold) shower would also help in other ways.
(Score: 0) by Anonymous Coward on Wednesday March 27 2019, @04:15PM (4 children)
Bender: Hey, brobot, what's your serial number?
Flexo: 3370318.
Bender: No way! Mine's 2716057!
[Bender and Flexo laugh. Fry starts to, and then ...]
Fry: I don't get it.
Bender: We're both expressible as the sum of two cubes!
(Score: 2) by Bot on Wednesday March 27 2019, @04:43PM (3 children)
Obviously you haven't noticed the glaring coincidence.
33 is also the name of the anthem of Italian mountain soldiers. Who have a pretty big association. Which is one century old THIS VERY YEAR. CPUblowing huh?
Account abandoned.
(Score: 3, Informative) by istartedi on Wednesday March 27 2019, @07:29PM (2 children)
The US 18th Amendment was passed in 1919, which enabled Prohibition the following year. It was later repealed in... 1933! The Amendment that repealed it? The 21st, which is the current drinking age. For a number of years, 18 was also a common drinking age.
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(Score: 3, Funny) by Bot on Wednesday March 27 2019, @11:35PM (1 child)
>the current drinking age
which oughta be... thirstythree!
Account abandoned.
(Score: 2) by istartedi on Thursday March 28 2019, @05:13PM
You should definitely be off mother's milk by then. Bartender, give me an apple juice... in a dirty sippy cup.
.
Appended to the end of comments you post. Max: 120 chars.
(Score: 2) by Snow on Wednesday March 27 2019, @10:11PM (2 children)
Why is this a 'Much studied equation'? Is there nothing else to research in the field of mathematics? Is this just a make-work project?
(Score: 2) by Bot on Wednesday March 27 2019, @11:42PM
I have an explanation:
normal people ⊇ people on the wrong side of the autism spectrum ⊇ mathematicians
Account abandoned.
(Score: 0) by Anonymous Coward on Thursday March 28 2019, @03:59AM
As a practical matter, number theory is the basis for cryptography. But it goes far beyond that, and the Quanta article is well written. If it weren't for human curiosity, we'd still be in the stone age.
(Score: 2) by darkfeline on Thursday March 28 2019, @03:30AM
It all makes sense now, 42 = Life³ + (The Universe)³ + Everything³
The Question of what Life, The Universe, and Everything are is left as an exercise for the reader.
Join the SDF Public Access UNIX System today!
(Score: 0) by Anonymous Coward on Thursday March 28 2019, @06:58PM
That's not a cube, it's three numbers to the third which gives the number 33.