Diophantine Equation Solved for Last Holdout Between 0 and 100 - 42

Jcolvin2
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Diophantine Equation Solved for Last Holdout Between 0 and 100 - 42

#1

Post by Jcolvin2 » Mon Sep 16, 2019 12:13 am

http://news.mit.edu/2019/answer-life-un ... atics-0910

The answer to life, the universe, and everything
Mathematics researcher Drew Sutherland helps solve decades-old sum-of-three-cubes puzzle, with help from "The Hitchhiker's Guide to the Galaxy."
Sandi Miller | Department of Mathematics
September 10, 2019

***

This sum of three cubes puzzle, first set in 1954 at the University of Cambridge and known as the Diophantine Equation x3+y3+z3=k, challenged mathematicians to find solutions for numbers 1-100. With smaller numbers, this type of equation is easier to solve: for example, 29 could be written as 33 + 13 + 13, while 32 is unsolvable. All were eventually solved, or proved unsolvable, using various techniques and supercomputers, except for two numbers: 33 and 42.

Booker devised an ingenious algorithm and spent weeks on his university’s supercomputer when he recently came up with a solution for 33. But when he turned to solve for 42, Booker found that the computing needed was an order of magnitude higher and might be beyond his supercomputer’s capability. Booker says he received many offers of help to find the answer, but instead he turned to his friend Andrew "Drew" Sutherland, a principal research scientist in the Department of Mathematics. “He’s a world’s expert at this sort of thing,” Booker says.

***

“There is a single integer parameter, d, that determines a relatively small set of possibilities for x, y, and z such that the absolute value of z is below a chosen search bound B,” says Sutherland. “One then enumerates values for d and checks each of the possible x, y, z associated to d. In the attempt to crack 33, the search bound B was 1016, but this B turned out to be too small to crack 42; we instead used B = 1017 (1017 is 100 million billion).

Otherwise, the main difference between the search for 33 and the search for 42 would be the size of the search and the computer platform used. Thanks to a generous offer from UK-based Charity Engine, Booker and Sutherland were able to tap into the computing power from over 400,000 volunteers’ home PCs, all around the world, each of which was assigned a range of values for d. The computation on each PC runs in the background so the owner can still use their PC for other tasks.

***

The method of using Charity Engine is similar to part of the plot surrounding the number 42 in the "Hitchhiker" novel: After Deep Thought’s answer of 42 proves unsatisfying to the scientists, who don’t know the question it is meant to answer, the supercomputer decides to compute the Ultimate Question by building a supercomputer powered by Earth … in other words, employing a worldwide massively parallel computation platform.
“This is another reason I really liked running this computation on Charity Engine — we actually did use a planetary-scale computer to settle a longstanding open question whose answer is 42.”

***

Using the Charity Engine network is also more energy-efficient. “For the most part, we are using computational resources that would otherwise go to waste,” says Sutherland. “When you're sitting at your computer reading an email or working on a spreadsheet, you are using only a tiny fraction of the CPU resource available, and the Charity Engine application, which is based on the Berkeley Open Infrastructure for Network Computing (BOINC), takes advantage of this. As a result, the carbon footprint of this computation — related to the electricity our computations caused the PCs in the network to use above and beyond what they would have used, in any case — is lower than it would have been if we had used a supercomputer.”
Sutherland and Booker ran the computations over several months, but the final successful run was completed in just a few weeks. When the email from Charity Engine arrived, it provided the first solution to x3+y3+z3=42:

42 = (-80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3

***

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Re: Diophantine Equation Solved for Last Holdout Between 0 and 100 - 42

#2

Post by Foggy » Mon Sep 16, 2019 7:02 am

:like:

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Re: Diophantine Equation Solved for Last Holdout Between 0 and 100 - 42

#3

Post by Bill_G » Mon Sep 16, 2019 7:12 am

Cubing a large negative number to sum with two large positives to arrive at a small number. No wonder it took so much cpu horsepower to solve.

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Re: Diophantine Equation Solved for Last Holdout Between 0 and 100 - 42

#4

Post by Foggy » Mon Sep 16, 2019 7:28 am

That happened in the movie, too also. :?
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Re: Diophantine Equation Solved for Last Holdout Between 0 and 100 - 42

#5

Post by Suranis » Mon Sep 16, 2019 7:40 am

Since they mentioned the Hitchhikers guide, I was actually thinking yesterday about a scene in the TV series (written by Douglas Adams so its unofficially official)where they actually find out the question.

Basically, in the show they have crashed back on earth on the B ship, and they realise they are about to fuck up the calculations to find the question as the Middle managers are about to wipe out the real Humans, so they realise they are the last chance to find The question. So they start tossing letters down on a board, and it starts forming words! And after a bit the full glorious question is revealed! And the question of Life, the Universe and everything (to which the answer is 42) is...

"What is Seven multiplied by Nine?"

Which, the mathamaticians among you will realise, equals 63.

And thus they realise there is something fundamentally very wrong with the universe.

There's some clips of the TV series if you are interested. Does not have that scene though...

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Re: Diophantine Equation Solved for Last Holdout Between 0 and 100 - 42

#6

Post by Northland10 » Mon Sep 16, 2019 7:45 am

:towel:
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Re: Diophantine Equation Solved for Last Holdout Between 0 and 100 - 42

#7

Post by Sam the Centipede » Mon Sep 16, 2019 11:05 am

Off Topic
As Suranis brought up Douglas Adams…

I heard a sort of radio soirée with Adams chatting about HHGG (obviously an old program because Adams died young about twenty years ago). He said he chose 42 as just a dull number. Likewise, he chose the name Arthur Dent as a sort of Everyman name. But some years later Adams discovered that a university scholar had written a thesis about deep connections he found between HHGG Arthur Dent's perambulation through the universe and the work The plaine mans path-way to heauen : wherein euery man may clearly see, whether he shall be saued or damned : set forth dialogue wise, for the better understanding of the simple written around 1600 by an English Puritan cleric named Arthur Dent. The earlier Dent's work was apparently an inspiration for John Bunyan when writing The Pilgrim's Progress.

It amused Adams too because implausible coincidence plays a major part in the HHGG, especially as the functional principle of the Infinite Improbability Drive. Fractal coincidence!

No cubes were harmed in the writing of this post.

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Re: Diophantine Equation Solved for Last Holdout Between 0 and 100 - 42

#8

Post by Sterngard Friegen » Mon Sep 16, 2019 11:07 am

Is the result exact or is there rounding?

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Re: Diophantine Equation Solved for Last Holdout Between 0 and 100 - 42

#9

Post by RoadScholar » Mon Sep 16, 2019 11:41 am

I think it has to be exact because they're all integers.
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Re: Diophantine Equation Solved for Last Holdout Between 0 and 100 - 42

#10

Post by Sterngard Friegen » Mon Sep 16, 2019 12:57 pm

RoadScholar wrote:
Mon Sep 16, 2019 11:41 am
I think it has to be exact because they're all integers.
Do you have a calculator that can do the cubing and subtraction? I don't.

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Re: Diophantine Equation Solved for Last Holdout Between 0 and 100 - 42

#11

Post by PaulG » Mon Sep 16, 2019 1:12 pm

Sterngard Friegen wrote:
Mon Sep 16, 2019 12:57 pm
RoadScholar wrote:
Mon Sep 16, 2019 11:41 am
I think it has to be exact because they're all integers.
Do you have a calculator that can do the cubing and subtraction? I don't.
Is one of your descendants taking an introductory programming course? He (or she) could write a program to verify it. Wouldn't it be Douglas Adams-y if there was a typo?

Anyway, I can verify the "2". Just take the last digits of the three numbers, cube them and add them. The "4" I will leave to someone else.

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Re: Diophantine Equation Solved for Last Holdout Between 0 and 100 - 42

#12

Post by Jcolvin2 » Mon Sep 16, 2019 1:25 pm

I am curious how the mathematicians proved that there were no solutions to the Diophantine Equation for k = 32. I haven't followed up, and I assume that I wouldn't understand it anyway.

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Re: Diophantine Equation Solved for Last Holdout Between 0 and 100 - 42

#13

Post by Sam the Centipede » Mon Sep 16, 2019 1:52 pm

Jcolvin2 wrote:
Mon Sep 16, 2019 1:25 pm
I am curious how the mathematicians proved that there were no solutions to the Diophantine Equation for k = 32. I haven't followed up, and I assume that I wouldn't understand it anyway.
It almost certainly involves the phrase "elliptic equation" early on which I take as a "no way sonny, go play with your bricks" warning. :(

Number Theory has always been a closed book to me, except for basic stuff about divisibility. I never got the fascination with prime numbers. Is it interesting that both of 42's neighbors are primes? Primes are the dull numbers that don't even factor, yawn. I don't expect any mathematician to agree with that view!!

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Re: Diophantine Equation Solved for Last Holdout Between 0 and 100 - 42

#14

Post by Gregg » Mon Sep 16, 2019 2:23 pm

RoadScholar wrote:
Mon Sep 16, 2019 11:41 am
I think it has to be exact because they're all integers.

That is correct, its integers.
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Re: Diophantine Equation Solved for Last Holdout Between 0 and 100 - 42

#15

Post by RoadScholar » Mon Sep 16, 2019 2:41 pm

Right, and any integer raised to a whole-number power is always going to be an integer.
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Re: Diophantine Equation Solved for Last Holdout Between 0 and 100 - 42

#16

Post by MaineSkeptic » Mon Sep 16, 2019 2:54 pm

Sterngard Friegen wrote:
Mon Sep 16, 2019 12:57 pm
RoadScholar wrote:
Mon Sep 16, 2019 11:41 am
I think it has to be exact because they're all integers.
Do you have a calculator that can do the cubing and subtraction? I don't.
Just popping in to say:

Python 3 does not have an upper limit on the integers it can handle, so the Python shell provides easy and quick verification. Just remember to change the "^" symbols to "**"!
Python 3.7.4 (tags/v3.7.4:e09359112e, Jul 8 2019, 20:34:20) [MSC v.1916 64 bit (AMD64)] on win32
Type "help", "copyright", "credits" or "license()" for more information.
>>> (-80538738812075974)**3 + 80435758145817515**3 + 12602123297335631**3
42
>>>

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Re: Diophantine Equation Solved for Last Holdout Between 0 and 100 - 42

#17

Post by PaulG » Mon Sep 16, 2019 4:23 pm

Things like FORTRAN are why I got into the field. Things like Python are why I'm retired.

I know when I'm beat. I'm surprised someone hasn't done it in Excel.

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Re: Diophantine Equation Solved for Last Holdout Between 0 and 100 - 42

#18

Post by Maybenaut » Mon Sep 16, 2019 5:03 pm

This whole thread sound like Charlie Brown’s teacher in my head.
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Re: Diophantine Equation Solved for Last Holdout Between 0 and 100 - 42

#19

Post by Northland10 » Mon Sep 16, 2019 8:05 pm

Maybenaut wrote:
Mon Sep 16, 2019 5:03 pm
This whole thread sound like Charlie Brown’s teacher in my head.
I did well in math in school. My day job is in fundraising analytics. I know what Python is and have considered playing around with it. I am often told I'm being to technical. I am well aware that the ultimate answer is 42.

With all that, for me this thread is not far from what you describe. :rolleye:
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Re: Diophantine Equation Solved for Last Holdout Between 0 and 100 - 42

#20

Post by Northland10 » Mon Sep 16, 2019 8:31 pm

Karl shouldn't be the only one who should find his thread invaded/illustrated by music theater so maybe it's time for a bit of theater and Einstein, Einstein, Einstein, theater, theater, Einstein on the, the the the, Einstein on on on the beach.

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Re: Diophantine Equation Solved for Last Holdout Between 0 and 100 - 42

#21

Post by Sterngard Friegen » Mon Sep 16, 2019 9:59 pm

RoadScholar wrote:
Mon Sep 16, 2019 2:41 pm
Right, and any integer raised to a whole-number power is always going to be an integer.
Ah. I see that now. I was assuming fractions.

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Re: Diophantine Equation Solved for Last Holdout Between 0 and 100 - 42

#22

Post by Sam the Centipede » Tue Sep 17, 2019 2:42 am

Sterngard Friegen wrote:
Mon Sep 16, 2019 9:59 pm
RoadScholar wrote:
Mon Sep 16, 2019 2:41 pm
Right, and any integer raised to a whole-number power is always going to be an integer.
Ah. I see that now. I was assuming fractions.
Well, I guess that's not irrational. :bag:

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Re: Diophantine Equation Solved for Last Holdout Between 0 and 100 - 42

#23

Post by Sterngard Friegen » Tue Sep 17, 2019 3:48 am

Sam the Centipede wrote:
Tue Sep 17, 2019 2:42 am
Sterngard Friegen wrote:
Mon Sep 16, 2019 9:59 pm
RoadScholar wrote:
Mon Sep 16, 2019 2:41 pm
Right, and any integer raised to a whole-number power is always going to be an integer.
Ah. I see that now. I was assuming fractions.
Well, I guess that's not irrational. :bag:
Nor imaginary.

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Re: Diophantine Equation Solved for Last Holdout Between 0 and 100 - 42

#24

Post by Sam the Centipede » Tue Sep 17, 2019 5:26 am

As we're on a mathematical gig, I guess this digression is not too evil. Even though my mathematical ability is limited, I can usually see the beauty that mathematicians find, but I just don't care; I find the natural world much more interesting than numbers and equations.

But I remember just being wowed in my late teens when I learned of the so-called Euler Identity which links the key constants of computation. It's usually expressed as
ei π = –1
but even more elegantly as
ei π + 1 = 0
where e=2.71828… is the exponential constant (can't remember a better name, it's always just e), π=3.14159… is … yeah, you know what π is, circles an' stuff, and i=√–1 is the bizarre square root of minus one. So in the second form it ties together the five most fundamental constants of mathematics. Neat!

It becomes less magical if one glances at the simplest infinite series for sine and cosine trigonometric functions to see that they clearly share some genes with the exp exponential function and it drops out of the equation of simple harmonic motion (pendulums, vibrating violin strings, etc.). But it's still amazingly neat!

Bizarrely too, raising i to itself, ii or (√–1)√–1 ought to create some sort of super-imaginary number (yeah, I know R is closed) but it actually has a very ordinary real value (and many complex values too). I think it's about 0.7ish but I can't be bothered to calculate it now. I am confident that some smarter persons here won't resist the temptation to fill that hole!

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Re: Diophantine Equation Solved for Last Holdout Between 0 and 100 - 42

#25

Post by Bill_G » Tue Sep 17, 2019 7:48 am

Okay. *Now* I'm going to need diophantine, or tylenol, or something for my headache.

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