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## Computer Generates Largest Math Proof Ever At 200TB of Data (phys.org) 143

An anonymous reader quotes a report from Phys.Org: A trio of researchers has solved a single math problem by using a supercomputer to grind through over a trillion color combination possibilities, and in the process has generated the largest math proof ever -- the text of it is 200 terabytes in size. The math problem has been named the boolean Pythagorean Triples problem and was first proposed back in the 1980's by mathematician Ronald Graham. In looking at the Pythagorean formula: a^2 + b^2 = c^2, he asked, was it possible to label each a non-negative integer, either blue or red, such that no set of integers a, b and c were all the same color. To solve this problem the researchers applied the Cube-and-Conquer paradigm, which is a hybrid of the SAT method for hard problems. It uses both look-ahead techniques and CDCL solvers. They also did some of the math on their own ahead of giving it over to the computer, by using several techniques to pare down the number of choices the supercomputer would have to check, down to just one trillion (from 10^2,300). Still the 800 processor supercomputer ran for two days to crunch its way through to a solution. After all its work, and spitting out the huge data file, the computer proof showed that yes, it was possible to color the integers in multiple allowable ways -- but only up to 7,824 -- after that point, the answer became no. Is the proof really a proof if it does not answer why there is a cut-off point at 7,825, or even why the first stretch is possible? Does it really exist?
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## Computer Generates Largest Math Proof Ever At 200TB of Data

• #### Yes. (Score:5, Insightful)

on Monday May 30, 2016 @08:14PM (#52213857) Homepage Journal
Of course it's a proof. Who taught you what the word "proof" means? Perhaps you were looking for "satisfying answer," dear editor?
• #### Re: (Score:1)

by Anonymous Coward

The question was actually asked by Bob Yirka, the reporter of TFA on phys.org.

But yeah.

• #### Re:Yes. (Score:5, Interesting)

on Monday May 30, 2016 @09:04PM (#52214105)

Indeed. I'm rather confused by the editorial commentary. To put it in terms of the summary regarding a question of color, imagine if someone asked the question, "What color is the sky?" Conjecture: The sky is blue.

Proof? Point an instrument at the sky and measure the light coming from it. Looking at the spectrum of frequencies coming from the sky, it falls into a range of colors that humans would generally associate with "blue."

That's it -- you've "proved" what the color of the sky is, i.e., "blue."

TFS instead starts asking, "But WHY are the frequencies emitted from the sky in the range that qualifies as blue? Why aren't there other dominant frequencies? Why do they fall in a particular range? Have we really proved what color the sky is???"

These are all very interesting questions, but they are irrelevant to the fact that the sky IS blue and one can prove it by measuring the frequency and correlating it with what humans call "blue." Proofs aren't generally about "why," and in fact many concise "elegant" mathematical proofs may be completely non-intuitive about showing why they work -- nevertheless they are considered valid proofs.

• #### Re: (Score:2)

Because shorter wavelengths of light are preferentially scattered off the molecules in the earth's atmosphere (that's also why the sunset is orange - - when a light ray (low angled at sunset) passes thru enough atmosphere, most of the blue and green get scattered away). Why does light scatter that way? Because the molecules are much smaller than the wavelengths involved, but the closer the size ratio, the bigger the interaction. Why? There's probably some confusing equation in electrodynamics that expla
• #### Re:Yes. (Score:5, Funny)

on Monday May 30, 2016 @11:06PM (#52214561)

Dude, we all know that if you keep digging these "why" questions, you'll just end up with 42.

• #### Re: (Score:3)

42... A very finite number indeed.

And therein lies the problem - Is it possible to demonstrate a proof through brute force?

• #### Re: (Score:2)

"Is it possible to demonstrate a proof through brute force?"

Yes.

Prove that the number of unique permutations of the first 3 natural numbers greater than 0 is 6.

Proof:

1 2 3
1 3 2
2 1 3
2 3 1
3 1 2
3 2 1

QED

• #### Re: (Score:3)

Is it possible to demonstrate a proof through brute force?

For certain classes of proof, yes.

These classes are ones where the solution space is finite, in which case one can merely enumerate through all the possibilities. Another is where you're proving in general something is true - like in this case. In which case all you need to do is find one counter-example to prove the hypothesis wrong. Which happened here - they simply went through and enumerated the possibilities and checked each and every one of the

• #### Re: (Score:2)

42... A very finite number indeed.

And therein lies the problem - Is it possible to demonstrate a proof through brute force?

For classes of problems whose solution space are finite, then yes, it is possible, and in many cases, desirable. If the solution space has properties similar to a finite subset of, say, the natural numbers, then you could prove by induction. Depending on the situation, some of these problems are easier to prove via induction, and other times, by simply iterating over the problem space.

• #### Re: (Score:2)

I thought the answer was anything to not go through a bunch of color swatches to pick a new color for the living room...

• #### Re: (Score:2)

Why does light scatter that way? Because the molecules are much smaller than the wavelengths involved, but the closer the size ratio, the bigger the interaction.

Rayleigh scattering? I think it might actually be a bit more complicated than you think. [xkcd.com]

• #### Re: (Score:1)

That's it -- you've "proved" what the color of the sky is, i.e., "blue."

Perhaps you've proved the frequency of the color from the sky, but are you sure the definition of blue to include this particular frequency for this object is correct definition of blue? Perhaps not.

• #### Re: (Score:2, Insightful)

by Anonymous Coward

It is upon the asker to define what would qualify as blue. Otherwise the problem is underspecified.

• #### Mathematical Proof != Scientific Theory (Score:5, Interesting)

on Tuesday May 31, 2016 @02:28AM (#52215159) Journal

These are all very interesting questions, but they are irrelevant to the fact that the sky IS blue and one can prove it by measuring the frequency and correlating it with what humans call "blue."

This is not a mathematical proof but a scientific theory supported by evidence. A mathematical proof, if correct, is always and absolutely true. The major difference is that suppose I did your experiment at night, or at sunrise/sunset, or on a cloudy day? I could get red, black, white or grey for the colour of my sky. All you can do in science is take data, come up with a thoery to explain that data and then test the predictions of that theory under conditions where nobody has tested it before to see whether it works. In your case it is very easy to disprove the theory that the sky is blue.

In fact you can never really prove a scientific theory - all you can say is that it works in all the situations it has been tested under. That's good enough to be extremely useful and to advance our understanding about how the universe works but it is not the same thing as a mathematical proof. This is why scientists spend time confirming that existing theories work in new situations but you never hear of mathematicians checking the pythagorus theorem again to confirm that it still works with new right-angled triangles.

• #### Re: (Score:1)

This is not a mathematical proof but a scientific theory supported by evidence.

I knew somebody was going to come and start arguing about this nonsense. Look -- I was trying to make a rough analogy in common language, first of all, which should have been clear to anyone (I thought). OBVIOUSLY the statement "the sky is blue" is not true at night or when it's cloudy or whatever. Duh. Thanks, Captain Obvious.

A mathematical proof, if correct, is always and absolutely true. [snip] In fact you can never really prove a scientific theory - all you can say is that it works in all the situations it has been tested under.

I want you to think hard about this. This is a common statement, but if you actually spend some time interrogating what you're claiming, you'll realize there is no PRACTICAL diff

• #### Re: (Score:1)

by Anonymous Coward

Mathematician and former research scientist here. (Sorry, I always post AC just on principle, so you'll have to verify everything I say from other reputable sources, not an online reputation.)

This post was just asking for a rebuttal. And in fact, I've trimmed up my rebuttal to just this tl;dr version:

Math: Logically provable theorems which apply only to the set of rules they were written for. You cannot perform a mathematical "experiment." Proofs can be logically flawed by making invalid assumptions. You wi

• #### Re: (Score:2)

Look -- I was trying to make a rough analogy in common language...OBVIOUSLY the statement "the sky is blue" is not true at night or when it's cloudy or whatever. Duh. Thanks, Captain Obvious.

Sometimes when dealing with General Ignorance you have to be Captain Obvious although here apparently not obvious enough. I understood your rough analogy for what it was and was extending it to explain to you the difference between a scientific theory and a mathematical proof. It doesn't matter what scientific statement you make you can never prove it is always true under all circumstances. The best you can say is that every time you have tested it the statement has proved to be correct.

So, you're stuck with a couple of possibilities...

No actually you ar

• #### Re: (Score:2)

Sometimes when dealing with General Ignorance you have to be Captain Obvious although here apparently not obvious enough.

Woah -- sorry, but that's over the line. I admittedly was a bit rude in making fun of your post, and for that I would apologize. But I did not accuse you yourself of being an idiot. This is impolite and unwarranted. If you spend even a few minute reviewing the history of my posts here, you'd know that I have a deep background in the history and philosophy of science. So accusing me of "general ignorance" in this area is really being a jerk.

Nevertheless, I'm in a good-natured mood, so I'll respond a b

• #### 1980s all over again (Score:2)

We already went through this in the 1980s. The four color theorem [wikipedia.org] - the idea that you can draw any map using just 4 colors and no neighboring regions with shared edges will have the same color - resolves down to 1476 possible simplified map configurations. Appel and Haken wrote a computer program to check each of those configurations, and none were found to violate the theorem. Thus the theorem must be true.

Mathematicians initially rejected the idea that this constituted a mathematical proof for the r
• #### Re: (Score:3)

Fun fact: point scientific equipment up at the sky, and you'll find that it's actually violet. It only looks blue because the human eye is more sensitive to the colour ranges that we call blue than the colour ranges that we call violet. But I agree with your sentiment :).
• #### Re: (Score:2)

Yep, that's why I specified that one should look at the whole spectrum and distribution and compare it to what humans would characterize as "blue." Obviously many light sources emit a wide variety of frequencies, but humans just perceive one hue out of that mess. The association, as you note, is complex, but pretty well studied.
• #### Re: (Score:2)

Point an instrument at the sky and measure the light coming from it. Looking at the spectrum of frequencies coming from the sky, it falls into a range of colors that humans would generally associate with "blue."

Or, you know, just look at it and observe that it is blue.

• #### Re: (Score:2)

Indeed. I'm rather confused by the editorial commentary. To put it in terms of the summary regarding a question of color, imagine if someone asked the question, "What color is the sky?" Conjecture: The sky is blue.

Proof? Point an instrument at the sky and measure the light coming from it. Looking at the spectrum of frequencies coming from the sky, it falls into a range of colors that humans would generally associate with "blue."

That's it -- you've "proved" what the color of the sky is, i.e., "blue."

Which illustrates the naive idea of what a proof is: a bit confused. For one thing, mathematical proof does not work this way - what comes closest, conceptually, is a physical experiment, but it fails, in the sense that the purpose of an experiment in empirical science is not to prove something; at best you can hope to disprove, or if you fail convincingly in doing that, you can confirm you theory. But a proof it aint. Only mathemetics can provide positive proof of anything, and only because it restricts it

• #### Re: (Score:3, Interesting)

Well, let me take the devil's advocate position for a moment.

Yes, it proves the conjecture, but by relying upon a mechanical process you miss out on something that is required by humans to bring the problem within the grasp of our limited attention spans and working memory: insight. So we know the answer to the conjecture, but it doesn't advance our understanding of the problem and its related problems the way a human proof would.

So you could say it's proof of the conjecture in the sense that it's convinc

• #### Re:Yes. (Score:4)

on Monday May 30, 2016 @10:17PM (#52214363)

So you could say it's proof of the conjecture in the sense that it's convincing evidence of the truth of the conjecture. But it doesn't necessarily contribute to mathematical knowledge in the same way you'd expect a traditional proof to. So you could well look at it as a "proof" but in only a restricted sense.

I'm not sure how enumerating a finite set and demonstrating that a property holds for all its elements (or conversely, that it doesn't hold for some of its elements) doesn't constitute "a traditional proof". Dealing with finite cases by enumerating them seems perfectly traditional to me.

• #### Re: (Score:2)

While it does seem reasonable to treat 'proofs that are essentially incomprehensible except in summary' differently from ones that humans can actually use; it seems uncomfortably likely that most proofs, likely the overwhelming majority, are of the incomprehensible flavor with just a tiny little island of math that a suitably intelligent person could actually plow through. Beyond that, it seems reasonable to suspect that only a vanishingly small slice of the proofs that could, in principle, be generated in
• #### Re: (Score:2)

I'm not sure why "uses a computer" necessitates a new class. It seems to me that your argument applies to proof by exhaustion whether done by hand or computer. Also it is not unheard of to try to reach simpler proofs after constructing a complex one - the investigation doesn't have to stop here, but now we know the answer.

• #### Re: (Score:2)

Well, it's a continuum. Dividing a problem into a trillion cases and dividing them into two might be the same in principle, but not in practice.

Dividing something into countably infinite cases probably counts as even more of an "insight-based" proof, because then carrying out the proof by induction requires organizing those cases in some way that makes your argument convenient.

• #### Re: (Score:2)

How is a machine 'computed' proof any different from a human 'computed' proof?

• #### Re:Yes. (Score:5, Insightful)

on Monday May 30, 2016 @09:39PM (#52214235)

It's proof, but the problem is the measure of "largest math proof ever" is dumb. I could let a computer (or preferably a cluster) generate proof that every natural number below 200 trillion is followed by another, and there are no gaps, and it would easily trump that as the "largest math proof ever". What's that you say, it's not the simplest proof? True, but my algorithm just didn't hit on the simplest proof yet... Or if you prefer, I can generate proof of the exact number of primes below 200 trillion, it would beat that record by far and as far as currently known, have no simpler proof. For that matter, the Great Internet Mersenne Prime Search is constantly generating proofs that, if written and dumped out sequentially, would beat the pants off this record. But I hope we're not (or shouldn't be) merely competing for "the largest waste of computing power ever" :)

• #### Re: (Score:2)

Also.... Let's say you just wanted to publish a really long proof, but it turned out what you wanted to prove wasn't true after all....... So you conveniently stripped out the one counterexample you found from the data.

How the hell is a human going to discover that you cheated, or that your proof is wrong because of a rounding error in your program?

With a normal proof your reviewers don't need to go through extreme measures to be able to verify your result.

• #### Re: (Score:2)

They discover you cheated by running the program again and seeing different results?
• #### Re: Yes. (Score:2)

One should never bet against dishonesty; but math is one of those places where 'X holds except for value N' is likely to be considered way more interesting than quietly suppressing the N case and declaring 'X holds'. The follow-up research on "why value N?" is almost certain to be of greater interest, and keep you getting cited longer, than the "yup, X is proven, move along."
• #### Re: (Score:1)

I wonder what the power bill was for the proof.

• #### Re:Yes. (Score:4, Informative)

on Monday May 30, 2016 @10:57PM (#52214531)
IBM Blue Ice supercomputer 325.40 kW x 48hours x 0.10/kwh = \$1,561.92.
• #### Re: Yes. (Score:1)

by Anonymous Coward

Well, that proof at least is less than a terabyte!

• #### Re: (Score:1)

Came here to say the same thing. The nice thing about a compact proof is that it may generalize to other situations or offer greater insights. This is certainly not a compact proof. But, to say it's not a proof is ludicrous. It's a very explicit and detailed proof.

It's the difference between adding up the numbers 1 through 100 sequentially (perhaps by counting on your fingers even), and using Gauss' insight to take a short cut. [wbilljohnson.com] The computer didn't take any insight-yielding shortcuts, but still got the

• #### Re: (Score:3)

Of course it's a proof.

Not so fast. There are several [umd.edu] mathematicians [arxiv.org] that don't necessarily agree [google.lv] with that statement. At least not in its strongest form. (We'll ignore the editor basing their argument on the wrong thing, of course being "satisfying" to a human has little bearing on the "proofness" of a mathematical argument).

In fact, whether computer generated proofs that are too large to check by human mathematicians are really proofs at all is a question that's alive (if not exactly well) in the mathematical community, and you

• #### Re: (Score:3)

On the other side of that coin is what happened to tables of integrals when computer algebra software came out. The tables of integrals in the back of calculus books were considered tablets passed down from on high. Ok exaggerating, but they were considered generally accurate except for typos. When computer algebra software became powerful enough to check them, they were found to be riddled with errors. When checked by hand the computer algebra software was found to correct.

So what's the difference? In the

• #### Re: (Score:2)

Yes. Don't take my argument to mean that I want to throw out the baby with the bathwater. I'm a firm believer in the use of computers in mathematics. I think we're only just scraping the surface of what's possible. (To add to your example: there's interesting work in statistics for example, where Monte Carlo type simulations have demonstrated that certain statistical tests are much more accurate and valuable, compared to others, on a wide range of different data. Results that were very difficult, if not imp

• #### Re: (Score:2)

No, a proof is a proof. What kind of a proof? It's a proof. A proof is a proof, and when you have a good proof, it's because it's proven.
• #### Very disappointing. (Score:2)

I really wanted the answer to be 42.

• #### Re: (Score:1)

I really wanted the answer to be 42.

They never finished the work: If you add all the digits in 7824 you get 21; the multiply by 2 (the number to which A, B, and c are raised), you get 42! Any real galactic hitchhiker can see this!

• #### Proof? (Score:3)

on Monday May 30, 2016 @08:18PM (#52213885)

This doesn't sound like a proof to me. It sound like they disproved the hypothesis at 7,825. It's kind of like saying that they proved that there are no strings of 8 consecutive zeros in the decimal representation of pi. Well, until you get up to about 172,000,000 digits, at which point there is one.

• #### Re:Proof? (Score:5, Informative)

on Monday May 30, 2016 @08:26PM (#52213913)

They proved that in every partition of the positive integers into two classes, one class contains a solution to the equation \$a^2+b^2 = c^2\$. The method of proof is by showing this is already two for any partition of the interval {1,2,...,7,825} into two classes.

This is not entirely surprising; probably there will eventually be quantitative bounds showing that if you colour the integers in {1,2,...,N} in two colours then there are at least f(N) monochromatic Pythagorean triples for some increasing function f(N). Then 7,825 is the first N where f(N)>0, that's all.

I do agree with you that Graham probably expected a proof of the quantitative type rather than a computer search, because many other Ramsey theory problems have quantitative solutions, but there's nothing wrong with starting with a computer search.

• #### Re: (Score:1)

So, how does this result in 200 TB? Does it represent the space required to store every digit of every calculation in a brute-force proof? If so, then maybe that''s the correct number. But that size does not make it "not human-verifiable." The algorithm used to generate it is probably less than 500 kB, something that any number of competent mathematicians could do.
• #### Re: (Score:2)

Agreed. I am curious if it could have been much smaller, perhaps a single page, to show just the set of values that failed the test, or was that what took 200 TB? Sounds like you'd need to show that all 7825 integers had been shown to have been cycled through all possible color combinations, but were all those combinations tabulated to be the 200 TB of data?

• #### Re: (Score:3)

Isn't it more like they *disproved* the hypothesis that "there are no strings of 8 consecutive zeros in pi" by finding an example at a given place?

As a non-mathematician - do I understand this correctly?
Expressing a proof that partitioning is possible for {1,2,...n} is simpler - you just provide an example. The hard part is proving that no such partition exists for a given n.
In this case, they both proved the specific cases up to 7,824, and disproved the general case by proving it impossible for the specif

• #### Science vs Maths. (Score:2)

Isn't it more like they *disproved* the hypothesis....[snip]....As a non-mathematician - do I understand this correctly?

Yes and no, terms such as "proof" are often used in a casual way. Science doesn't offer proof or truth, it offers evidence. Maths doesn't have hypotheses, it has conjectures and truth. This is because Maths is an axiomatic system, the universe is not an axiomatic system therefore Science doesn't have axioms, it has assumptions. A properly formed mathematical conjecture (or statement) is "proven" by demonstrating it is true or false by the axioms of the system.

Having said that, the conjecture was in inde

• #### Re: (Score:2)

Maths doesn't have hypotheses, it has conjectures and truth. This is because Maths is an axiomatic system

The axioms are the hypothesis. Change them and you get a different mathematics. In that sense, in mathematics everything is hypothetical.

Actually mathematics does have hypotheses. However there are very few, they are called axioms and hardly raise any doubt about their validity, with the possible exception of the axiom of choice.

And Euclid's fifth postulate [wikipedia.org]: change this hypothesis and you get different geometry [wikipedia.org].

There are other issues with axioms:

Do we restrict ourselves to first order predicate logic (like Tarski's Axioms for Euclidean geometry [wikipedia.org]) or do we allow higher order logic (like Hilbert's axioms for Euclidean geometry [wikipedia.org])? Such choices have consequences for completeness and decidability.

• #### THIS belongs on Slashdot (Score:3, Insightful)

on Monday May 30, 2016 @08:18PM (#52213887)
This is absolutely news for nerds. Please post stories like this one and fewer "Why Trump/Clinton are lizard people" stories.

On topic, what is the content of this 200tb proof? Is that just a text file where each character is a bit? How many libraries of congress is this proof?

Whatever the content, congrats to this team of mathematicians.
• #### Contents of the 200TB proof (Score:3)

On topic, what is the content of this 200tb proof? Is that just a text file where each character is a bit? How many libraries of congress is this proof?

Consider the integers between 1 and 7285. To each integer i assign a boolean variable P_i. These variables encode the partition of these integers into two classes (think of P_i as encoding the statement ""the integer i belongs to the first class").

Now let \$a,b,c\$ be a Pythagorean triple (a^2+b^2=c^2 and 1

Finally, the claim "every triple is not monochrom

• #### Re:Contents of the 200TB proof (Score:5, Insightful)

on Monday May 30, 2016 @08:55PM (#52214039)
Sorry, the system ate my "less than" signs. Here's a corrected version.
1. Consider the integers between 1 and 7285. To each integer i assign a boolean variable P_i. These variables encode the partition of these integers into two classes (think of P_i as encoding the statement ""the integer i belongs to the first class").
2. Now let \$a,b,c\$ be a Pythagorean triple (a^2+b^2=c^2 and each is between 1 and 7285). Construct the boolean expression Q_{a,b,c} = (P_a & P_b & !P_c) || (P_a & !P_b & P_c) .." (disjunction of 6 clauses each being a conjunction of three terms) describing the 6 ways in which a triple can be non-monochromatic). So Q_{a,b,c} encodes the assertion "the integers a,b,c are not in the same class" by writing out the 6 ways in which a,b,c can belong to two classes without all belonging to the same class).
3. Finally, the claim "every triple is not monochromatic" is obtained by taking the conjuction of the Q_{a,b,c} over all triples (a,b,c) as above. It's a huge boolean expression and the goal is to show that it always evaluates to FALSE (in other words, that for any boolean assignment to all the P_i), the huge conjugation always takes the values.
4. The proof works by manipulating this boolean expression: it has 200TB of instructions on how to manipulate this huge boolean expressions step-by-step in ways that obviously don't change its truth value, so that at the end one of the clauses in the conjunction simply reads "FALSE", making the whole expression indeed universally false. A computer program discovered this list of manipulations, and a separate (much simpler) program can easily verify that the manipulations are of the right kind (they don't change the truth value) and that at the end of the manipulation you get a clause saying FALSE.
• #### Re: (Score:3)

You can use less than and greater than signs if you HTML-escape them, eg: "&lt;" (<) and "&gt;" (>).

• #### Re: (Score:2, Insightful)

This is absolutely news for nerds.

I'm going to play devil's advocate and point out that this story literally boils down to "someone used a computer to do what computers do."

• #### Of course it's a proof! (Score:2)

It's just that it's an experimental proof instead of a purely mathematical one.

And it's hardly the first time this has been done - the 4 color math problem was proved by computation back in 1976.

• #### Computer-assisted proofs are proofs (Score:3)

on Monday May 30, 2016 @08:19PM (#52213895)

Well, part of the argument is proving that (if implemented correctly) the algorithm actually solves the problem. But in fact this part is redundant -- because what the computer does is actually write out a proof of the theorem.

The point is that while coming up with a proof takes work or ingenuity, verifying the correctness of an proposed proof (written in sufficient detail) is purely mechanical. In other words, you don't need to believe or check anything the researchers have done. You simply need to take the output of their program and use a proof-checker to verify that this output is a valid proof.

For many combinatorial theorems this is the way of the future, and while the submitted may be unsatisfied by a proof which doesn't provide intuition, isn't that better than no proof at all?

• #### Re: (Score:3)

Also, consider simple proofs by induction. Besides the actual induction, you need to show that a trivial case works, which is usually plain arithmetic that could be done on a computer. More complicated proofs may require multiple brute-force cases before the actual math can be done. For example, I recall a proof of Bertrand's postulate which first needs individual cases for n < 2000.

It's a bit like physics where you need some concrete system of measurements, a real-world grounding for your abstract wo

• #### Writing this out by hand would have been a b*tch (Score:2)

Brings back memories of hand writing proofs in Adv Calc and Analytical Calc, getting 6 pages in and finding a mistake. 199.99999 TB written by hand and finding a mistake = bad year :-)
• #### Retarded question ... (Score:2)

Is the proof really a proof if it does not answer why there is a cut-off point at 7,825, or even why the first stretch is possible? Does it really exist?
When you have counted all possible outcomes and put a note for each one into a basket, collecting all notes of the same kind, then the result is a proof.

What else should it be? There is no "why" needed.

The "why" would be interesting as it might lead to a formula where you just put in a few parameters and get a "colour" as answer.

The easiest proof btw is the

• #### Limits (Score:2)

Is the proof really a proof if it does not answer why there is a cut-off point at 7,825, or even why the first stretch is possible? Does it really exist?

• #### Clobbering With Statistics (Score:1)

It's just "Clobbering With Statistics" which is fun, and probably easier to get funding for, as you just buy a bunch of equipment.

• #### One question: (Score:2)

Has anyone checked all 200TB of data? Are you certain that there is no flaw in the output?
• #### QED (Score:2)

If the file ends with "QED", then but definition it is a proof.

• #### Re: (Score:2)

If the file ends with "QED", then b[y] definition it is a proof.

Except of course if it is a text about Quantum Electro-Dynamics.

• #### entire [N]/y (Score:1)

So it says it is not entire? [Y]/n
• #### Proof of cutoff point (Score:1)

One could image the editors being just a little bit 'happier' with the proof if just a bit more information was provided about the number 7825.

Correct me if I'm wrong, but 7825 has to be part of at least one Pythagorean Triple, no? If you take all the integers up to 7824 and you can divide them up, but then you fail when you add 7825, then 7825 has to be part of a triple, otherwise it wouldn't be a tipping point.

So there has to be at least one set of numbers a and b such that a + b = 7825. a and b must be s

• #### The proof program (Score:1)

# Claim: It is not possible to make a partition of the
# natural numbers into to sets so that
# _all_ pythagorean tripples are split up,
# having some elements in one partition and
# some in the other.
#
# The math proof relies on this being true for all triples
# of numbers up to N = 7285, something which can be tested by
# a computer.
#
# We create a logic expression that is true iff
# there exists some partition such that all triples
# are split, and false otherwise. The computer
# checks that this is indeed al
• #### Re: (Score:1)

#The range should not include 0 - so much for trying to be concise...

range(1,N+1)
• #### simple proof (Score:2)

I've got a simple proof of this but it's slightly too large for this comment box. I'll post it laterrrrrrrrrrrrr...
• #### Is this really a proof? (Score:1)

Is this really a proof? I was always taught that a proof shows that the argument presented is always true -- not just for the first 7,824 times as is the case here, but always (obviously given the axioms used).

For instance, in Euclidean geometry, one can prove that the sum of the angles of a triangle are two 90 degree angles or 180 degrees. They don't total to 180 degrees up to some finite point as in the article.

Don't get me wrong. There was impressive work done here. It just doesn't amount to a proof, un

• #### What would Ramanujan Do? (Score:3)

on Tuesday May 31, 2016 @10:09AM (#52216655) Journal
What Would Ramanujan [wikipedia.org]

Do?

He will simply calculate all possible Pythagorean triples in his head, write down 7285 in a piece of paper as the "solution" and leave the proof as an elementary exercise to the reader.

• #### Cumulative Rounding Error (Score:2)

That's all I got.

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