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mikejuk writes "A proof [preprint PDF] has been proposed for the Collatz conjecture about hailstone sequences. A hailstone sequence starts from any positive integer n the next number in the sequence is n/2 if n is even and 3n+1 if n is odd. The conjecture is that this simple sequence always ends in one. Simple to state but very difficult to prove and it has taken more than 60 years to get close to a solution."

Re:... and someone finds a fault in the proof. (2, Interesting)

Anonymous Coward | more than 3 years ago | (#36342754)

Ok, I'm not a mathematician but when I read:

"What's odd is that this assertion looks on the surface extremely simple. In particular, it is true for any 2l which is divisible by 4, and is true for any 2l I can reasonably imagine... but in the cathedral of mathematics experimental evidence is no evidence at all.

Disclaimer: I am a physics graduate student, not an established mathematician by any means. This is the best I can grok the paper."

I wonder... perhaps the paper's author also thought it was simple and obvious and possibly even proved elsewhere?

Anyway. Isn't refuting the proof, just claiming it is incomplete. And from what I recall, mathematicians often submit proofs others consider "incomplete" since they kinda assume those things are obvious.

Re:... and someone finds a fault in the proof. (1)

There's a difference between types of incompleteness in proofs. The first type is when details are left to the reader, which can occur for a variety of reasons: it's too tedious to completely write up; they needed the paper to be shorter; they didn't want to lose sight of their main goal by getting lost in details; etc. This type is pretty innocuous, though it can make papers denser than they otherwise need to be. The second type is when the author doesn't notice the source of incompleteness. A good classical example of this case is Gauss' first proof of the fundamental theorem of arithmetic. He essentially used the Jordan curve theorem in his proof. That theorem happens to be intuitively obvious but wasn't proved for decades after Gauss' proof. This gap wasn't spotted for years, though it wasn't horrific as far as gaps go.

My guess (from reading a few independent sources that point out the same issue) is that this assertion falls into the second type of incompleteness, and that it's a particularly horrible form of it in that it assumes a reformulated version of the desired conclusion.

In any case, let's stop making stories about preprints proving/falsifying famous conjectures. After they've been reviewed by some experts, great, make a story--just not until then.

Re:... and someone finds a fault in the proof. (1)

In any case, let's stop making stories about preprints proving/falsifying famous conjectures. After they've been reviewed by some experts, great, make a story--just not until then.

This isn't the Washington Post or the New York Times. I don't think Slashdot needs to have the same journalistic standards as publications "of record." I don't think there's any harm in reporting these kinds of stories. Personally, I think it's pretty cool to see these kinds of attempts, even if they turn out to be WAY off base. And it's not like this story had a sensational headline--all it said was that a proof was "proposed." Anyway, that's my two cents.

Re:... and someone finds a fault in the proof. (1)

I suppose my issue is that pretty much every day someone somewhere comes up with a "proof" of a famous conjecture. Most of these are patently nuts, but some are (on the surface, at least) reasonably serious. It seems a bit strange to single out some of these for special attention. It's also almost always a letdown to read these types of stories. "Oh, cool, it was solved? I wonder how they did--oh wait, the first high rated comment says there's a big hole." In this headline I disliked the subtitle, "Hailstone Sequences End In 1". It seems like it should have ended in a question mark. As it is, it's a statement equivalent to saying the proof is correct, which I initially took to mean that the proof was very likely correct (as opposed to some preprint that doesn't seem to have been reviewed very well).

That said, I can understand people enjoying these types of articles even if they're based on something completely wrong. I suppose I just want something a little different and more rigorous out of/. news.

Re:... and someone finds a fault in the proof. (1)

Here, I think, one can comment on how this business of science is changing markedly.

I think if this conjecture was proved a decade or so ago, we would know of it's existence, and of its proving and the surprisingly hard road to its proof in classrooms; just like we see the problem of findings median in linear time. But today, as we speak, we are in fact commenting on a blog entry which refers to the proof and are discussing the type of flaw it may have; even though we do in the end yield the matter to experts.

And this is not the only instance. The talk of Arsenic based life form [slashdot.org] , or, more recently, the hints of Higgs Boson at LHC [slashdot.org] among other things that I have come to know of on slashdot, which has enough readership to have an effect [wikipedia.org] named after it. And slashdot is usually has a relay link to another blog.

Amazing.

Re:... and someone finds a fault in the proof. (3, Insightful)

Anonymous Coward | more than 3 years ago | (#36342756)

That's not a fault in the strict sense, that's an omission. Besides, this IS a preprint - it's not been published, hasn't even been refereed or peer-reviewed yet, so it's not in the least bit surprising it's not perfect.

Remember Wiles' proof of FLT? That one went through quite a few iterations, and it took years until all the flaws were ironed out - big ones, in some cases, that required significant changes to the whole thing. But in the end, the proof stood.

Re:... and someone finds a fault in the proof. (1)

I don't think it's remotely fair to compare this paper to the FLT proof. For one thing, the content of this one is around 11 pages (the PDF includes many pages of tables at the end) while Wiles' final proof is over 100 pages. This is much more "elementary", as well. I don't remember the source, but I remember hearing that at the time it was made the number of people qualified to comment on Wiles' proof could fit in a small room, which is certainly not true of this proof. Wiles invented a graduate course where he secretly presented his proof before announcing it publicly, whereas this doesn't have nearly that amount of content. [That's not to say this proof can't possibly be correct, just that it's a vastly different case than FLT.]

Re:... and someone finds a fault in the proof. (0)

Anonymous Coward | more than 3 years ago | (#36344074)

This is much more "elementary", as well. I don't remember the source, but I remember hearing that at the time it was made the number of people qualified to comment on Wiles' proof could fit in a small room

Hi, grandparent AC responding.

Wiles' proof was indeed very difficult to digest, but that is because Wiles actually developed a lot of new mathematics to tackle the problem - how would anyone else be supposed to be able to understand it without studying it first? As you may recall, he even wrote a second paper to go along with the one on FLT, titled, if I recall correctly, "ring-theoretic properties of certain Hecke algebras". It was full of technical stuff that he needed, and quite long in itself. Together, I recall the papers ran at... 270 pages or so? A *lot*, in any case.

You're reading too much into my above comment, however. I only meant that it's normal for papers to be tweaked, proofs corrected and holes filled, and cited Wiles' paper as a well-known example. I could easily have picked another, more obscure one, but this is one that I imagined people would actually be familiar with.

Re:... and someone finds a fault in the proof. (1)

The actual hailstone sequence definition says that if a 1 is reached, the sequence ends. The article submitter omitted that, probably to troll slashdot.

The actual hailstone sequence definition says that if a 1 is reached, the sequence ends. The article submitter omitted that, probably to troll slashdot.

And it worked.

Re:neat (0)

Anonymous Coward | more than 3 years ago | (#36343810)

He was probably too busy writing a shitty run-on sentence: "A hailstone sequence starts from any positive integer n [some punctuation or something should go here, you mong] the next number in the sequence is n/2 if n is even and 3n+1 if n is odd."

That very nice (-1, Troll)

Anonymous Coward | more than 3 years ago | (#36342750)

But he still won't get laid.

Re:That very nice (0)

Anonymous Coward | more than 3 years ago | (#36344106)

The sequence ends in 1 rather than 1, 4, 2, 1, 4, 2..... by definition. A hailstone sequence has one additional rule, which is that the first 1 is the last 1, and the sequence ends.

Re:before too many question this (0)

Anonymous Coward | more than 3 years ago | (#36343462)

Leave it to mathematicians to choose to add that third rule, rather than simply conjecturing "the sequence eventually reaches 1".

Those are actually orthogonal. For example, in your version the sequence is infinite, whereas with the additional rule it is probably bounded, and the length might even be amenable to calculation from the input.

Re:before too many question this (0)

Anonymous Coward | more than 3 years ago | (#36344236)

I was wondering about that. Glad I checked to see if it was addressed before I asked.;-)

A hailstone sequence starts from any positive integer n the next number in the sequence is n/2 if n is even and 3n+1 if n is odd.

It wouldn't have taken any more time to properly punctuate this "sentence" once -- for everyone -- than it takes everyone to punctuate it in their heads in order to make sense of it. I realize they just cut and pasted the bulleted points -- minus the bullets -- but c'mon, they didn't put those there just for decoration.

You shouldn't feel the need to apologize. The people who are presumably being paid to act in an editorial capacity for Slashdot are the individuals who should be taking the time to ensure submissions are readable. They're clearly willing and able to edit submissions, yet have rarely shown any inclination toward putting effort into anything beyond adding subjective comments or making questionable changes that result in submitters having to point out what the editors have done.

This has been a problem for as long as I can remember, and while it's excusable from a volunteer-run operation, a paying for-profit operation should be willing to invest in someone willing to do basic proofreading and fact-checking while approving reader-submitted links, favoured weblogger glurge, and Slashvertisements.

Then again, I don't think much of the hypothesis that financial profit is an obligatory motivator for competence, so I'm not surprised that garbled sentences go right by the staff; hiring someone to check the spelling and grammar would cut into profits, after all, and as long as people keep paying for crap, what motivation is there to change?

I first heard about this conjecture many years ago (25?) and did what most geeks would do; I wrote a program to test all odd integer values and check that they did in fact reach 1.

The obvious approach is to remember the count for each value as you try them, then check any intermediate result against this table: If found, just add that value to the current count and store the result.

This approach breaks down when you want to test starting values much larger than 2^32, because the space to store the table becomes larger than available memory.

One of the first optimizations is to realize that the result of taking 3n+1 starting from an odd (n) will always be even, so you can simply include the following n/2 (a shift) and increase the count, while getting rid of the multiplication by 3:

Odd(n):
n = 2m+1
3n+1 = 6m + 4
(3n+1)/2 = 3m + 2 = n + (n>>1) + 1

In asm this can be simplified to:

again:
shr eax,1;; Assume even, so divide by two and shift the bottom bit into carry
jz done;; If the result of the shift was zero, we're done!

inc edx;; Increment number of steps
jnc again;; No carry means even input, so go back up

inc edx;; One extra increment;; The starting value was odd, so now we need to multiply by 3 and add 2:
lea eax,[eax+eax*2+2]
jmp again

done:

A much more sophisticated step is to see that the next N operations are determined by the bottom N bits of the current value (as long as there are more than N bits available), basically allowing you to convert those N operations into a multiplication, an add and a shift right.

Next you realize that the intermediate values can very quickly overflow a 32-bit unsigned integer, and even using 64-bit values does not get you very far.

My approach back in those days was to use a variable number of 32-bit counters:

With a single counter I check for getting to 1 or overflowing so I need to use two counters.

With two (or more) counters I test for the top counter becoming zero, allowing me to reduce the number of counters, or for overflow forcing another incease.

All of this is of course much easier in asm than C,since you can use the ADD/ADC combination to perform arbitrary precision

And this is where you lose the thread. From the point of mathematic proof, "as many odd integers as possible" has *nothing* to do with "all odd integers" unless you stumble upon one that proves it false. Finding any finite number of odd integers for which it is true accomplishes nothing towards proving it true for all odd integers.

Re:Easy to state, fun to program! (0)

Anonymous Coward | more than 3 years ago | (#36344076)

I wrote a program to test all odd integer values

ALL odd integer values? Where'd you buy your computer? I gotta get my next one from there!

This site [alanturing.com] has absolutely no information on where to buy one, but according to this one [logic.at] the GP's program might not stop soon.

Re:Easy to state, fun to program! (0)

Anonymous Coward | more than 3 years ago | (#36344156)

It's true, there are only a finite amount of odd numbers. That's really important to modern mathematics as well. That's why we keep searching for gigantic prime numbers. Because, soon enough, we will have found them all. Once we have those we can construct the remaining even numbers. Once our number line has been completed, we can finally factor out some of the digits of pi and eventually get that thing to terminate. The future shall be a wondrous place.

I know mathematicians study these mathematics for reasons entirely unrelated to any practical applications of them. And I know that most of the value in proofs and understanding is totally independent of any applications, as the applications' utility are typically of relatively brief lifetime while the math is eternal.

But I'm not a mathematician, nor am I immortal. I'm an engineer. So I'm always interested in whether these proofs (or new insights) have actual immediate or reachable applications. These are called "hailstone" sequences - is the math any use in weather or geology? Or any other?

Well, it's a great example of a simple algorithm which may or may not halt. So it's useful for teaching algorithms, compiler design, and the halting problem (which is enormously useful).

But assuming it does always end in 1... no, there's no use for the proof. The proof (or it's techniques) might lead to more useful applications though.

Re:Applications? (1)

Anonymous Coward | more than 3 years ago | (#36342958)

But I'm not a mathematician, nor am I immortal. I'm an engineer. So I'm always interested in whether these proofs (or new insights) have actual immediate or reachable applications

Excuse me but aren't engineers supposed to be taking math and turning it into something useful. You sound more like a user - only using what is already there.

Re:Applications? (1)

Anonymous Coward | more than 3 years ago | (#36343078)

Well, that is pretty much what engineering is. You would be surprised on how little math is actually used in engineering. Perhaps you are thinking about physicist that occationally have to do some major advances in math to be able to solve their problems. Notable examples are Newton and his integrals and Green and his theorem. The math developed by pure mathematicians are seldom of use outside their field but still have an entertainment value for those in the field.

No, engineers apply known applications to problems they have to solve. Which is why we want newly known applications. And why those of us with problems for which there are no known applications for solutions are interested in the possibility of new applications when the bottleneck at the point of understanding has been passed.

Lots of us "take math" who aren't mathematicians. The vast majority of us, in fact.

Hey look buddy, I'm an Engineer, that means I solve problems. Not problems like 'What is beauty?', because that would fall within the purview of your conundrums of philosophy.

I solve practical problems.

For instance: How am I going to stop some big mean mother hubbard from tearing me a structurally superfluous new behind?

The answer: Use a gun. And if that don't work? Use more gun.

No immediate or even forseeable applications (and it has nothing to do with hailstones). But the guys who were working on number theory in the early 1900's (Hardy for one, who specifically refused to work on any maths that had real-world applications) didn't forsee the future uses in cryptography...

The reason mathematicians are interested in the 3n+1 problem is that we do not have a very good understanding of the process -- it is a fairly simple process to describe, but it is not so easy to explain why it would always fall into the same cycle (1,4,2,1,4,2,1,4,2). A lot of problems in number theory are like this; Fermat's last theorem was similarly easy to state but very difficult to understand and prove. The real-life application of these problems is often more related to the various methods required to solve them than the problem itself.

For example, consider the problem of the quintic formula. As everyone with a middle school education should know, there is a formula that gives the solution to any linear equation (ax+b = 0), and there is a formula that gives the solutions to any quadratic equation (ax^2 + bx + c = 0). Some more educated people will also know that there are formulas for cubic equations (ax^3 + bx^2 + cx + d = 0) and quartic equations (ax^4 + bx^3 + cx^2 + dx + e = 0). The obvious question is, "What about quintic equations (ax^5 + bx^4 + cx^3 + dx^2 + ex + f = 0)?"

The answer is a somewhat intellectually interesting, "No, there is no formula that gives the solution to all quintic equations (using only arithmetic and radicals)." There is no real-world application of that answer; we can get good enough approximations of the solutions to quintic equations by various numeric methods, which is perfectly fine for any problem that involves solving a quintic. However, the proof that there is no quintic formula involves fields of mathematics that are very much applicable to real-world problems: group theory and field theory are very important in cryptography and certain branches of physics. Additionally, those fields of study led to the research of more general abstract algebra, with still more real-world application.

So, no, there is no real-world use for the Collatz conjecture, at least none that I am aware of. In all likelihood, the proof of the Collatz conjecture will lead to some practical application, or a better understanding of certain real world problems.

Offtopic: Is there any theory of solving hyperoperation [wikipedia.org] equations generally, or any theoretical work on why some types of hyperoperation equations are solvable by their respective formula but some aren't?

But I'm not a mathematician, nor am I immortal. I'm an engineer. So I'm always interested in whether these proofs (or new insights) have actual immediate or reachable applications. These are called "hailstone" sequences - is the math any use in weather or geology? Or any other?

I'm sure someone will call you a math philistine or something like that for your question, but I took it seriously. And as I read it, I thought "sure there must be. I bet a quick Google search would turn up some." But the only practical use I turned up was in the creation of graphs, charts and illustrations of the sequences themselves.

Oh, I think I misconstrued the person I was replying to. They were probably referring to applications of this purported proof (which I don't believe is correct) while I was referring to mathematicians' work on the Collatz conjecture, or the conjecture itself and numerous related topics and generalizations.

These simple proofs are like Debian - by itself it's unconfigured so it won't play proprietary stuff. But they can be combined to form other results.

As a superfast dirty guess, there's a computing application here somewhere. The conjecture itself deals with binary division, and the +1 part of the other rule has something to do with a power of 2 result being one more than an odd number next to it.

Fast guesses for apps have to do with lowbyte-highbyte number encoding, etc as well as a bunch of other stuff. (Failsafes in fuzzy-AI feel like they are in there somewhere too.)

Re:Applications? (0)

Anonymous Coward | more than 3 years ago | (#36345592)

The proof itself may be useful, but the theorem (once proven) is unlikely to be - if it were, then we could assume the conjecture and apply it sort of like we do with the Riemann Hypothesis. It has been empirically verified for "reasonable" numbers, and so almost any application that relied on the theorem is fine for practical purposes. Knuth might be deeply offended by people doing so, but baring its use in things like nuclear control systems, it would be fine.

The reasons why the theorem may matter is that the notion of a "reasonable" number may not hold up to the quantities of throughput that happen at chip speeds. Think of stack carry-through register operations. On a 64 core chip when all that integrates it may not hold at 10^16 or something with a race condition threatened.

"Hailstone" doesn't seem to have anything whatsoever to do with actual weather or geology. MathWorld's [wolfram.com] take on it:

Such sequences are called hailstone sequences because the values typically rise and fall, somewhat analogously to a hailstone inside a cloud.
While a hailstone eventually becomes so heavy that it falls to ground, every starting positive integer ever tested has produced a hailstone sequence that eventually drops down to the number 1 and then "bounces" into the small loop 4, 2, 1,....

It lists as one of its sources this book [amazon.com] on the etymology of math terms in English. It looks interesting. Maybe I'll get a copy myself....

I had a use for a reverse "etymological mathematics dictionary" just this past week. I was trying to collect from people working on projects the "base projects" in the dependency network of all their projects. I was looking for the complementary word to "dependent" in the dependencies. After a while I discovered "pendent [wikipedia.org] ", which is the thing on which a dependency depends, from geometry.

But I don't see myself ever again buying a printed dictionary of any subject, except for collectible sake. Because I need hyperlinks and searching in any reference document for it to be useful. Maybe a bathroom book to open at "random", though that would disadvantage terms at the extreme ends of the alphabet. And soon enough ruin a $50 purchase.

If anything the concepts necessary to reason about the problem are probably useful, even if the problem itself is not. Depending on the way the numbers spread in the algorithm, it may have applications for cryptography, or hashes and digests. Algorithms that either generate numbers with greater range from a simpler number, or calculate a simpler number from one of a greater range.

It is undoubtedly also possible to make datastructures and algorithms, that uses the sequence, though there is no guarantee it will be an improvement (compare to Fibonacci numbers for instance). Mmmmm... Hailstone heap. It may not be very good, but it would have one badass name. Hail Stoneheap.

Understanding if a sequence ends is very important. You don't like it when your word processor crashes (ends) when you try to save a file do you?

The halting problem is unsolvable, but we have to solve it anyway, as well as we can. And that's just one possible application. The fact is, you have to have a paintbrush before you can paint, and you have to develop the proof before you can say what the proof technique can be used for.

If mathematicians had slowed down every time some "practically minded person" (I'm trying to be nice here) asked if there was a practical application to being able to solve a certain problem, we'd have absolutely no modern technology. None at all. Cell phone, gps, microwave, electricity coming to your house-- I can't even begin to describe the magnitude of the loss we'd have.

Even more than that, it seems extremely odd that a person would ask what the "application" of humans being able to solve a math problem could be. What the hell can be more important to you than humanity taking 1 step forward and being able to reason more effectively?

Nobody asked mathematicians to slow down. And what's "extremely odd" is that you tried ranking what's more important with an obnoxious question after I explained why I'm interested in an application if it exists. What the hell is wrong with wanting an application in addition to humanity taking 1 step forward?

If you can't understand that from what I posted, I expect you have absolutely nothing to offer in helping solve the hailstone sequence. So just shut up. You're holding humanity back with your infinitesimal contribution.

1*3 +1 = 4 (0)

Anonymous Coward | more than 3 years ago | (#36342934)

if it reaches 1 then the next would be 4 , so how could it possibly end in 1 ?

It seems that the field of mathematics is cut throat. Just the tone alone from that blog post would make me reconsider even trying to break into mathematics as a researcher. I thought only computer geeks are like that on USENET.

If you (metaphorically) go on the podium with your fly open and your dick hanging out, you deserve to be laughed off the stage.

I am waaay behind in my mathematics (since university), but even to me the paper looked rather fishy. If a problem has not been solved for decades, a prudent researcher would not push a paper like this out. And he surely wouldnt give interviews in the press talking how good he is and how he only recently started working on the problem...

Re:Debunked already (0)

Anonymous Coward | more than 3 years ago | (#36346398)

As a mathematician I do see this sort of back and forth exchanges, sometimes they are explicitly malicious, and other times they are more tactful. Mathematicians (myself included) often have a high standard of what they consider quality mathematics. You see debates over proof techniques, 'correct' hierarchy of ideas, etc. Very much like how programmers argue over stylistic issues, even when the results are the same (e.g. a program that does the same thing).

This has made some very critical thinking and meticulous people, if issues of presentation won't escape them, then certainly a flaw in an argument will not either. It is frustrating to hear public claims of proof of this or proof of that and have the presentation be not just flawed, but be so muddled that it takes considerable effort to discover that flaw (Remember the P = NP guy months ago? Same thing).

My mentor from when I was an undergraduate would put it this way: "If you can't teach it, you don't know it." Maybe an overstatement, but if you write a paper claiming something, then it should be so crystal clear that the possibility of error is unthinkable. You should be able to practically grok the argument on a first or second read. That's good mathematics. A proof is really you explaining why something is true. Just like in a college class you expect a lecture to make sense and you rate the quality of the professor on how well he explains something, you should also expect this from a proof.

Mathematics deals in certainties. An integer is odd (or prime, or divisble, or constant, or whatever), or it's not, by a very strict definition.

The problem stems from *proving* that. The mathematician (i.e. someone who studied mathematics at, say, degree level or higher) that can actually write a proof is surprisingly rare. The one that can write a perfect, undeniable, accepted, doubt-free proof in a single hit is would be seen as an absolute genius better than any other mathematician that's ever lived - more "clever" than Einstein, Hawking, etc. Because we're human and we always miss *something* in any proof ever offered for even the simplest of things.

Most proofs are like security programs on computers - they'll have holes with virtually certainty, and it's only accepted as "proof" when the holes have been found, poked at for 20+ years and every holes patched, countered and fixed. That's the process called "peer review". Look at any mathematical-based security process in computer science and you see the same pattern - cryptography, clever exploits, chroot-breaking etc. It takes decades to find MOST of the holes and an infinite amount of time to find all of them.

Any "proof" that comes from someone who hasn't been poking holes in others papers, or endlessly patching their proof for years based on the input of humongously talented third-parties is, history shows, 100% bullshit. Just read the story behind the solution to Fermat's Last Theorem, for example.

The problem is that the mathematical equivalent of "free energy" fanatics, etc., are people who claim to have proven something and yet have zero experience of providing actual proofs that are accepted by the mathematical community. It's elitest, sure, but it helps to filter out the nutters if you just ignore EVERYTHING posted on arxiv.org until it appears in a reputable, third-party-reviewed journal and has had ten rebuttals, all of which have been neatly countered using fabulously complex arguments.

You can now "prove" the four-colour theorem in an afternoon with a home computer and some mathematical knowledge. But it took *decades* to convince the mathematical community that a proof which relies almost entirely on the output of a computer to perform the grunt work was acceptable. And it was still in doubt from 1976 up until around 2005 as to whether it was actually a "proof" or not.

If someone claims to have solved a long-standing mathematical problem, with zero previous experience, treat them how you'd treat someone who claims that they can double the output of your car engine without using any extra energy source if you let them take the car apart. Yes, technically and historically, such things have been possible by certain marvellous innovations in the field. But would you really let him tinker with your car on the basis of that without knowledge of that person himself?

P=NP is a field that attracts people just like the "free/perpetual energy/motion" nutters, for instance. There's been at least three or four slashdot stories of "proofs" of that since the beginning of the year. None of them came to anything, all of them were firmly rebutted, and none of the authors were heard of again. It's not a general hostility in mathematics - just towards the time-wasters and nutters that think A-level maths qualifies them to provide a definitive proof of something new in an afternoon.

Re:Debunked already (0)

Anonymous Coward | more than 3 years ago | (#36348846)

Read the "about me" on that blog, he is a CS guy not a mathematician.

Hailstone sequence (0)

Anonymous Coward | more than 3 years ago | (#36343866)

Are these the hailstones that are the size of flat cylindrical objects (like coins) rather than the size of spherical objects (peas, golf balls' softballs ?

This sequence will always collapse to 1 when we reach any number that is represented by 2^x. That is to say that when we reach a number whose prime factors are only twos. The entire purpose of 3n+1 is to manipulate the previous number to try to get a number that can be represented by 2^x. Dividing all even numbers by two is just a simple way to test for 2^x. Given enough iterations we will eventually reach a number that can be represented by 2^x. Of course this is only what I concluded years ago and I could be wrong.

...which is itself inevitable once (n mod 4) = 1. (Stupid slashdot filters out ≅ symbol)

The problem really becomes proving that for any {n: (n mod 4) = 2,3}, the sequence will eventually yield {n: (n mod 4) = 1}.

2^n (0)

Anonymous Coward | more than 3 years ago | (#36346868)

Well, it seems you will always ultimately land on a number which is a power of 2. At that point, you'll be doing nothing but dividing by 2. You'll only be n steps from 1, with no escape.

The conclusion of the conjecture seems obvious. Showing that the sequence *must* at some point arrive at a power of 2, I'm sure, is quite hard.

I'm not a math major, and I have very little experience with any proof. I've never written one. But I would first think this is a 3 part problem. One of n=even, n=odd and n=2^x. Proving all will become 2^x seems to be the goal.

Any even integer n divided by 2 is an integer half the value Any odd integer n multiplied by 3 is an odd integer. Any odd integer n plus one is an even integer

Through induction, any power of two in this sequence will end in 1. That is to say if n_0 is 2^x, then the sequence will always be even and always end in 1. 2/2 =1, 8/2/2/2=1.

As an infinite sequence, n will always become odd for any even n_0, not a power of 2.

Since odd n always have 1 added, they will eventually become a power of 2.

Looking at the partial Hailstone tree [hubrisarts.com] as a network, it's interesting that the tree can fork only at every other node, even for nodes that aren't on the "power of 2" limb. There's no case in which forks occur in neighboring nodes.

That would be a useful property to have in practical situations such as the generation of clock trees in digital integrated circuits. It would be interesting to see if similar rules could be applied to CAD systems.

## ... and someone finds a fault in the proof. (2)

## Musically_ut (1054312) | more than 3 years ago | (#36342680)

## Re:... and someone finds a fault in the proof. (2, Interesting)

## Anonymous Coward | more than 3 years ago | (#36342754)

Ok, I'm not a mathematician but when I read:

"What's odd is that this assertion looks on the surface extremely simple. In particular, it is true for any 2l which is divisible by 4, and is true for any 2l I can reasonably imagine... but in the cathedral of mathematics experimental evidence is no evidence at all.

Disclaimer: I am a physics graduate student, not an established mathematician by any means. This is the best I can grok the paper."

I wonder... perhaps the paper's author also thought it was simple and obvious and possibly even proved elsewhere?

Anyway. Isn't refuting the proof, just claiming it is incomplete.

And from what I recall, mathematicians often submit proofs others consider "incomplete" since they kinda assume those things are obvious.

## Re:... and someone finds a fault in the proof. (1)

## FrootLoops (1817694) | more than 3 years ago | (#36343264)

There's a difference between types of incompleteness in proofs. The first type is when details are left to the reader, which can occur for a variety of reasons: it's too tedious to completely write up; they needed the paper to be shorter; they didn't want to lose sight of their main goal by getting lost in details; etc. This type is pretty innocuous, though it can make papers denser than they otherwise need to be. The second type is when the author doesn't notice the source of incompleteness. A good classical example of this case is Gauss' first proof of the fundamental theorem of arithmetic. He essentially used the Jordan curve theorem in his proof. That theorem happens to be intuitively obvious but wasn't proved for decades after Gauss' proof. This gap wasn't spotted for years, though it wasn't horrific as far as gaps go.

My guess (from reading a few independent sources that point out the same issue) is that this assertion falls into the second type of incompleteness, and that it's a particularly horrible form of it in that it assumes a reformulated version of the desired conclusion.

In any case, let's stop making stories about preprints proving/falsifying famous conjectures. After they've been reviewed by some experts, great, make a story--just not until then.

## Re:... and someone finds a fault in the proof. (1)

## Opyros (1153335) | more than 3 years ago | (#36343634)

Was this supposed to be "the fundamental theorem of algebra"? But I agree with your points.

## Re:... and someone finds a fault in the proof. (2)

## barlevg (2111272) | more than 3 years ago | (#36344298)

In any case, let's stop making stories about preprints proving/falsifying famous conjectures. After they've been reviewed by some experts, great, make a story--just not until then.

This isn't the Washington Post or the New York Times. I don't think Slashdot needs to have the same journalistic standards as publications "of record." I don't think there's any harm in reporting these kinds of stories. Personally, I think it's pretty cool to see these kinds of attempts, even if they turn out to be WAY off base. And it's not like this story had a sensational headline--all it said was that a proof was "proposed." Anyway, that's my two cents.

## Re:... and someone finds a fault in the proof. (1)

## FrootLoops (1817694) | more than 3 years ago | (#36346164)

I suppose my issue is that pretty much every day someone somewhere comes up with a "proof" of a famous conjecture. Most of these are patently nuts, but some are (on the surface, at least) reasonably serious. It seems a bit strange to single out some of these for special attention. It's also almost always a letdown to read these types of stories. "Oh, cool, it was solved? I wonder how they did--oh wait, the first high rated comment says there's a big hole." In this headline I disliked the subtitle, "Hailstone Sequences End In 1". It seems like it should have ended in a question mark. As it is, it's a statement equivalent to saying the proof is correct, which I initially took to mean that the proof was very likely correct (as opposed to some preprint that doesn't seem to have been reviewed very well).

That said, I can understand people enjoying these types of articles even if they're based on something completely wrong. I suppose I just want something a little different and more rigorous out of /. news.

## Re:... and someone finds a fault in the proof. (1)

## Musically_ut (1054312) | more than 3 years ago | (#36345162)

businessof science is changing markedly.I think if this conjecture was proved a decade or so ago, we would know of it's existence, and of its proving and the surprisingly hard road to its proof in classrooms; just like we see the problem of findings median in linear time. But today, as we speak, we are in fact

commentingon ablogentry which refers to the proof and are discussing thetypeof flaw it may have; even though we do in the end yield the matter to experts.And this is not the only instance. The talk of Arsenic based life form [slashdot.org] , or, more recently, the hints of Higgs Boson at LHC [slashdot.org] among other things that I have come to know of

onslashdot, which has enough readership to have an effect [wikipedia.org] named after it. And slashdot is usually has a relay link to another blog.Amazing.

## Re:... and someone finds a fault in the proof. (3, Insightful)

## Anonymous Coward | more than 3 years ago | (#36342756)

That's not a fault in the strict sense, that's an omission. Besides, this IS a preprint - it's not been published, hasn't even been refereed or peer-reviewed yet, so it's not in the least bit surprising it's not perfect.

Remember Wiles' proof of FLT? That one went through quite a few iterations, and it took years until all the flaws were ironed out - big ones, in some cases, that required significant changes to the whole thing. But in the end, the proof stood.

## Re:... and someone finds a fault in the proof. (1)

## FrootLoops (1817694) | more than 3 years ago | (#36343358)

## Re:... and someone finds a fault in the proof. (0)

## Anonymous Coward | more than 3 years ago | (#36344074)

This is much more "elementary", as well. I don't remember the source, but I remember hearing that at the time it was made the number of people qualified to comment on Wiles' proof could fit in a small room

Hi, grandparent AC responding.

Wiles' proof was indeed very difficult to digest, but that is because Wiles actually developed a lot of new mathematics to tackle the problem - how would anyone else be supposed to be able to understand it without studying it first? As you may recall, he even wrote a second paper to go along with the one on FLT, titled, if I recall correctly, "ring-theoretic properties of certain Hecke algebras". It was full of technical stuff that he needed, and quite long in itself. Together, I recall the papers ran at... 270 pages or so? A *lot*, in any case.

You're reading too much into my above comment, however. I only meant that it's normal for papers to be tweaked, proofs corrected and holes filled, and cited Wiles' paper as a well-known example. I could easily have picked another, more obscure one, but this is one that I imagined people would actually be familiar with.

## Re:... and someone finds a fault in the proof. (1)

## ObsessiveMathsFreak (773371) | more than 3 years ago | (#36343252)

Oh, the ignominy!

To have one's proof faulted is unfortunate; to have it faulted on the internet is simply mortifying.

## Re:... and someone finds a fault in the proof. (0)

## Anonymous Coward | more than 3 years ago | (#36343562)

You underestimate us. We are lejoin

## not reviewed yet? (-1)

## Anonymous Coward | more than 3 years ago | (#36342684)

## neat (0)

## Anonymous Coward | more than 3 years ago | (#36342726)

I propose for some numbers it ends in a repeating loop of 2, 1, 4, 2, 1, 4

## Re:neat (1)

## cela0811 (1901860) | more than 3 years ago | (#36342844)

## Re:neat (1)

## multisync (218450) | more than 3 years ago | (#36342890)

One is an odd number, so you multiply it by three and add one for four, which is even, so you divide by two ... and it's turtles all the way down ...

## Re:neat (1)

## Rob the Bold (788862) | more than 3 years ago | (#36342950)

One is an odd number, so you multiply it by three and add one for four, which is even, so you divide by two ... and it's turtles all the way down ...

The conjecture as stated in the summary, in addition to being poorly punctuated, is not "official". Or complete.

## Re:neat (1)

## Surt (22457) | more than 3 years ago | (#36343100)

Except the actual sequence definition declares that the sequence ends at the first instance of one in the sequence.

## Re:neat (1)

## multisync (218450) | more than 3 years ago | (#36344054)

You're right. One is the defined end of the sequence.

## Re:neat (1)

## Surt (22457) | more than 3 years ago | (#36342896)

The actual hailstone sequence definition says that if a 1 is reached, the sequence ends. The article submitter omitted that, probably to troll slashdot.

## Re:neat (1)

## Rob the Bold (788862) | more than 3 years ago | (#36343050)

The actual hailstone sequence definition says that if a 1 is reached, the sequence ends. The article submitter omitted that, probably to troll slashdot.

And it worked.

## Re:neat (0)

## Anonymous Coward | more than 3 years ago | (#36343810)

He was probably too busy writing a shitty run-on sentence: "A hailstone sequence starts from any positive integer n [some punctuation or something should go here, you mong] the next number in the sequence is n/2 if n is even and 3n+1 if n is odd."

## That very nice (-1, Troll)

## Anonymous Coward | more than 3 years ago | (#36342750)

But he still won't get laid.

## Re:That very nice (0)

## Anonymous Coward | more than 3 years ago | (#36344106)

## Summary incorrect! (0)

## Anonymous Coward | more than 3 years ago | (#36342874)

## before too many question this (3, Informative)

## Surt (22457) | more than 3 years ago | (#36342904)

The sequence ends in 1 rather than 1, 4, 2, 1, 4, 2..... by definition. A hailstone sequence has one additional rule, which is that the first 1 is the last 1, and the sequence ends.

## Re:before too many question this (0)

## Anonymous Coward | more than 3 years ago | (#36343462)

Leave it to mathematicians to choose to add that third rule, rather than simply conjecturing "the sequence eventually reaches 1".

## Re:before too many question this (1)

## Surt (22457) | more than 3 years ago | (#36343504)

Those are actually orthogonal. For example, in your version the sequence is infinite, whereas with the additional rule it is probably bounded, and the length might even be amenable to calculation from the input.

## Re:before too many question this (0)

## Anonymous Coward | more than 3 years ago | (#36344236)

I was wondering about that. Glad I checked to see if it was addressed before I asked. ;-)

## So in other words... (1)

## LordRobin (983231) | more than 3 years ago | (#36346774)

...the conjecture could be more simply stated "Every hailstone sequence ends".

Or even more simply stated "Every hailstone sequence eventually contains a power of 2".

------RM

## Easy to state, hard to read. (3, Insightful)

## Rob the Bold (788862) | more than 3 years ago | (#36342910)

A hailstone sequence starts from any positive integer n the next number in the sequence is n/2 if n is even and 3n+1 if n is odd.

It wouldn't have taken any more time to properly punctuate this "sentence" once -- for everyone -- than it takes everyone to punctuate it in their heads in order to make sense of it. I realize they just cut and pasted the bulleted points -- minus the bullets -- but c'mon, they didn't put those there just for decoration.

## Re:Easy to state, hard to read. (1)

## mikejuk (1801200) | more than 3 years ago | (#36343010)

## Isn't that what "editors" are for? (1)

## Platinum Dragon (34829) | more than 3 years ago | (#36344508)

You shouldn't feel the need to apologize. The people who are presumably being paid to act in an editorial capacity for Slashdot are the individuals who should be taking the time to ensure submissions are readable. They're clearly willing and able to edit submissions, yet have rarely shown any inclination toward putting effort into anything beyond adding subjective comments or making questionable changes that result in submitters having to point out what the editors have done.

This has been a problem for as long as I can remember, and while it's excusable from a volunteer-run operation, a paying for-profit operation should be willing to invest in someone willing to do basic proofreading and fact-checking while approving reader-submitted links, favoured weblogger glurge, and Slashvertisements.

Then again, I don't think much of the hypothesis that financial profit is an obligatory motivator for competence, so I'm not surprised that garbled sentences go right by the staff; hiring someone to check the spelling and grammar would cut into profits, after all, and as long as people keep paying for crap, what motivation is there to change?

## Easy to state, fun to program! (4, Interesting)

## Terje Mathisen (128806) | more than 3 years ago | (#36343688)

I first heard about this conjecture many years ago (25?) and did what most geeks would do; I wrote a program to test all odd integer values and check that they did in fact reach 1.

The obvious approach is to remember the count for each value as you try them, then check any intermediate result against this table:

If found, just add that value to the current count and store the result.

This approach breaks down when you want to test starting values much larger than 2^32, because the space to store the table becomes larger than available memory.

One of the first optimizations is to realize that the result of taking 3n+1 starting from an odd (n) will always be even, so you can simply include the following n/2 (a shift) and increase the count, while getting rid of the multiplication by 3:

Odd(n):

n = 2m+1

3n+1 = 6m + 4

(3n+1)/2 = 3m + 2 = n + (n>>1) + 1

In asm this can be simplified to:

again: ;; Assume even, so divide by two and shift the bottom bit into carry ;; If the result of the shift was zero, we're done!

shr eax,1

jz done

inc edx ;; Increment number of steps ;; No carry means even input, so go back up

jnc again

inc edx ;; One extra increment ;; The starting value was odd, so now we need to multiply by 3 and add 2:

lea eax,[eax+eax*2+2]

jmp again

done:

A much more sophisticated step is to see that the next N operations are determined by the bottom N bits of the current value (as long as there are more than N bits available), basically allowing you to convert those N operations into a multiplication, an add and a shift right.

Next you realize that the intermediate values can very quickly overflow a 32-bit unsigned integer, and even using 64-bit values does not get you very far.

My approach back in those days was to use a variable number of 32-bit counters:

With a single counter I check for getting to 1 or overflowing so I need to use two counters.

With two (or more) counters I test for the top counter becoming zero, allowing me to reduce the number of counters, or for overflow forcing another incease.

All of this is of course much easier in asm than C,since you can use the ADD/ADC combination to perform arbitrary precision

Terje

## Re:Easy to state, fun to program! (1)

## Chris Mattern (191822) | more than 3 years ago | (#36343822)

ALL odd integer values? Where'd you buy your computer? I gotta get my next one from there!

## Re:Easy to state, fun to program! (1)

## Terje Mathisen (128806) | more than 3 years ago | (#36344046)

Well, doing it for ALL values is what the mathematicians are trying to do, right?

I hope it was clear from my post that I meant "as many odd integers as possible"! :-)

Terje

## Re:Easy to state, fun to program! (1)

## Unequivocal (155957) | more than 3 years ago | (#36344724)

It was obvious - and a nice write up. Thanks. That guy's a troll or a pedant.

## Re:Easy to state, fun to program! (1)

## Chris Mattern (191822) | more than 3 years ago | (#36349358)

And this is where you lose the thread. From the point of mathematic proof, "as many odd integers as possible" has *nothing* to do with "all odd integers" unless you stumble upon one that proves it false. Finding any finite number of odd integers for which it is true accomplishes nothing towards proving it true for all odd integers.

## Re:Easy to state, fun to program! (0)

## Anonymous Coward | more than 3 years ago | (#36344076)

ALL odd integer values? Where'd you buy your computer? I gotta get my next one from there!

This site [alanturing.com] has absolutely no information on where to buy one, but according to this one [logic.at] the GP's program might not stop soon.

## Re:Easy to state, fun to program! (0)

## Anonymous Coward | more than 3 years ago | (#36344156)

## Re:Easy to state, fun to program! (1)

## FatLittleMonkey (1341387) | more than 3 years ago | (#36346264)

## Re:Easy to state, fun to program! (1)

## FatLittleMonkey (1341387) | more than 3 years ago | (#36346360)

Never mind... http://hubrisarts.com/collatz.png [hubrisarts.com]

of the

number'syou searched<Sigh>

## Re:Easy to state, fun to program! (1)

## Terje Mathisen (128806) | more than 3 years ago | (#36347974)

Working with starting points less than 1e9 you can try 670617279 which leads to a sequence of 987 steps.

Starting with 319804831 gets you close to 64-bit overflow, with an intermediate value of 1414236446719942480.

Terje

## Re:Easy to state, hard to read. (1)

## alexo (9335) | more than 3 years ago | (#36353900)

That depends. Were you referring to the bullets or to the /. editors?

## Applications? (1)

## Doc Ruby (173196) | more than 3 years ago | (#36342920)

I know mathematicians study these mathematics for reasons entirely unrelated to any practical applications of them. And I know that most of the value in proofs and understanding is totally independent of any applications, as the applications' utility are typically of relatively brief lifetime while the math is eternal.

But I'm not a mathematician, nor am I immortal. I'm an engineer. So I'm always interested in whether these proofs (or new insights) have actual immediate or reachable applications. These are called "hailstone" sequences - is the math any use in weather or geology? Or any other?

## Re:Applications? (2)

## nedlohs (1335013) | more than 3 years ago | (#36342938)

No.

## Re:Applications? (1)

## wisty (1335733) | more than 3 years ago | (#36342968)

Well, it's a great example of a simple algorithm which may or may not halt. So it's useful for teaching algorithms, compiler design, and the halting problem (which is enormously useful).

But assuming it does always end in 1 ... no, there's no use for the proof. The proof (or it's techniques) might lead to more useful applications though.

## Re:Applications? (1)

## Anonymous Coward | more than 3 years ago | (#36342958)

Excuse me but aren't engineers supposed to be taking math and turning it into something useful. You sound more like a user - only using what is already there.

## Re:Applications? (1)

## Anonymous Coward | more than 3 years ago | (#36343078)

Well, that is pretty much what engineering is. You would be surprised on how little math is actually used in engineering.

Perhaps you are thinking about physicist that occationally have to do some major advances in math to be able to solve their problems. Notable examples are Newton and his integrals and Green and his theorem.

The math developed by pure mathematicians are seldom of use outside their field but still have an entertainment value for those in the field.

## Re:Applications? (1)

## Doc Ruby (173196) | more than 3 years ago | (#36343258)

No, engineers apply known applications to problems they have to solve. Which is why we want newly known applications. And why those of us with problems for which there are no known applications for solutions are interested in the possibility of new applications when the bottleneck at the point of understanding has been passed.

Lots of us "take math" who aren't mathematicians. The vast majority of us, in fact.

## Re:Applications? (1)

## larry bagina (561269) | more than 3 years ago | (#36343724)

Hey look buddy, I'm an Engineer, that means I solve problems. Not problems like 'What is beauty?', because that would fall within the purview of your conundrums of philosophy.

I solve practical problems.

For instance: How am I going to stop some big mean mother hubbard from tearing me a structurally superfluous new behind?

The answer: Use a gun. And if that don't work? Use more gun.

## Re:Applications? (3, Funny)

## Doc Ruby (173196) | more than 3 years ago | (#36344094)

If that's your solution, you're a terrible engineer.

## Re:Applications? (1)

## wierd_w (1375923) | more than 3 years ago | (#36346740)

Agreed! EVERYONE knows that GOOD engineers use duct tape instead!

(You can solve ANYTHING with well-used duct tape!)

## Re:Applications? (1)

## MasterPatricko (1414887) | more than 3 years ago | (#36342974)

No immediate or even forseeable applications (and it has nothing to do with hailstones). But the guys who were working on number theory in the early 1900's (Hardy for one, who specifically refused to work on any maths that had real-world applications) didn't forsee the future uses in cryptography ...

## Re:Applications? (5, Insightful)

## betterunixthanunix (980855) | more than 3 years ago | (#36343026)

For example, consider the problem of the quintic formula. As everyone with a middle school education should know, there is a formula that gives the solution to any linear equation (ax+b = 0), and there is a formula that gives the solutions to any quadratic equation (ax^2 + bx + c = 0). Some more educated people will also know that there are formulas for cubic equations (ax^3 + bx^2 + cx + d = 0) and quartic equations (ax^4 + bx^3 + cx^2 + dx + e = 0). The obvious question is, "What about quintic equations (ax^5 + bx^4 + cx^3 + dx^2 + ex + f = 0)?"

The answer is a somewhat intellectually interesting, "No, there is no formula that gives the solution to all quintic equations (using only arithmetic and radicals)." There is no real-world application of that answer; we can get good enough approximations of the solutions to quintic equations by various numeric methods, which is perfectly fine for any problem that involves solving a quintic. However, the proof that there is no quintic formula involves fields of mathematics that are very much applicable to real-world problems: group theory and field theory are very important in cryptography and certain branches of physics. Additionally, those fields of study led to the research of more general abstract algebra, with still more real-world application.

So, no, there is no real-world use for the Collatz conjecture, at least none that I am aware of. In all likelihood, the proof of the Collatz conjecture will lead to some practical application, or a better understanding of certain real world problems.

## Re:Applications? (1)

## Doc Ruby (173196) | more than 3 years ago | (#36343350)

Thanks for that explanation.

Offtopic: Is there any theory of solving hyperoperation [wikipedia.org] equations generally, or any theoretical work on why some types of hyperoperation equations are solvable by their respective formula but some aren't?

## Re:Applications? (1)

## Rob the Bold (788862) | more than 3 years ago | (#36343040)

But I'm not a mathematician, nor am I immortal. I'm an engineer. So I'm always interested in whether these proofs (or new insights) have actual immediate or reachable applications. These are called "hailstone" sequences - is the math any use in weather or geology? Or any other?

I'm sure someone will call you a math philistine or something like that for your question, but I took it seriously. And as I read it, I thought "sure there must be. I bet a quick Google search would turn up some." But the only practical use I turned up was in the creation of graphs, charts and illustrations of the sequences themselves.

Some are pretty.

## Re:Applications? (1)

## StripedCow (776465) | more than 3 years ago | (#36343274)

Sure. Its main application is an insomnia treatment for mathematicians.

## Re:Applications? (1)

## FrootLoops (1817694) | more than 3 years ago | (#36343396)

## Re:Applications? (1)

## cslax (1215816) | more than 3 years ago | (#36343756)

## Re:Applications? (1)

## FrootLoops (1817694) | more than 3 years ago | (#36343968)

## Re:Applications? (1)

## TaoPhoenix (980487) | more than 3 years ago | (#36343348)

These simple proofs are like Debian - by itself it's unconfigured so it won't play proprietary stuff. But they can be combined to form other results.

As a superfast dirty guess, there's a computing application here somewhere. The conjecture itself deals with binary division, and the +1 part of the other rule has something to do with a power of 2 result being one more than an odd number next to it.

Fast guesses for apps have to do with lowbyte-highbyte number encoding, etc as well as a bunch of other stuff. (Failsafes in fuzzy-AI feel like they are in there somewhere too.)

## Re:Applications? (0)

## Anonymous Coward | more than 3 years ago | (#36345592)

The proof itself may be useful, but the theorem (once proven) is unlikely to be - if it were, then we could assume the conjecture and apply it sort of like we do with the Riemann Hypothesis. It has been empirically verified for "reasonable" numbers, and so almost any application that relied on the theorem is fine for practical purposes. Knuth might be deeply offended by people doing so, but baring its use in things like nuclear control systems, it would be fine.

## Re:Applications? (1)

## TaoPhoenix (980487) | more than 3 years ago | (#36346012)

Hi AC.

The reasons why the theorem may matter is that the notion of a "reasonable" number may not hold up to the quantities of throughput that happen at chip speeds. Think of stack carry-through register operations. On a 64 core chip when all that integrates it may not hold at 10^16 or something with a race condition threatened.

## Re:Applications? (2)

## FrootLoops (1817694) | more than 3 years ago | (#36343470)

Such sequences are called hailstone sequences because the values typically rise and fall, somewhat analogously to a hailstone inside a cloud. While a hailstone eventually becomes so heavy that it falls to ground, every starting positive integer ever tested has produced a hailstone sequence that eventually drops down to the number 1 and then "bounces" into the small loop 4, 2, 1, ....

It lists as one of its sources this book [amazon.com] on the etymology of math terms in English. It looks interesting. Maybe I'll get a copy myself....

## Re:Applications? (1)

## Doc Ruby (173196) | more than 3 years ago | (#36344140)

Thanks for the explanation of the name.

I had a use for a reverse "etymological mathematics dictionary" just this past week. I was trying to collect from people working on projects the "base projects" in the dependency network of all their projects. I was looking for the complementary word to "dependent" in the dependencies. After a while I discovered "pendent [wikipedia.org] ", which is the thing on which a dependency depends, from geometry.

But I don't see myself ever again buying a printed dictionary of any subject, except for collectible sake. Because I need hyperlinks and searching in any reference document for it to be useful. Maybe a bathroom book to open at "random", though that would disadvantage terms at the extreme ends of the alphabet. And soon enough ruin a $50 purchase.

## Re:Applications? (1)

## Carewolf (581105) | more than 3 years ago | (#36344690)

If anything the concepts necessary to reason about the problem are probably useful, even if the problem itself is not. Depending on the way the numbers spread in the algorithm, it may have applications for cryptography, or hashes and digests. Algorithms that either generate numbers with greater range from a simpler number, or calculate a simpler number from one of a greater range.

It is undoubtedly also possible to make datastructures and algorithms, that uses the sequence, though there is no guarantee it will be an improvement (compare to Fibonacci numbers for instance). Mmmmm... Hailstone heap. It may not be very good, but it would have one badass name. Hail Stoneheap.

## Re:Applications? (1)

## EuclideanSilence (1968630) | more than 3 years ago | (#36347486)

The halting problem is unsolvable, but we have to solve it anyway, as well as we can. And that's just one possible application. The fact is, you have to have a paintbrush before you can paint, and you have to develop the proof before you can say what the proof technique can be used for.

If mathematicians had slowed down every time some "practically minded person" (I'm trying to be nice here) asked if there was a practical application to being able to solve a certain problem, we'd have absolutely no modern technology. None at all. Cell phone, gps, microwave, electricity coming to your house-- I can't even begin to describe the magnitude of the loss we'd have.

Even more than that, it seems extremely odd that a person would ask what the "application" of humans being able to solve a math problem could be. What the hell can be more important to you than humanity taking 1 step forward and being able to reason more effectively?

## Re:Applications? (1)

## Doc Ruby (173196) | more than 3 years ago | (#36348542)

Nobody asked mathematicians to slow down. And what's "extremely odd" is that you tried ranking what's more important with an obnoxious question after I explained why I'm interested in an application if it exists. What the hell is wrong with wanting an application

in additionto humanity taking 1 step forward?If you can't understand that from what I posted, I expect you have absolutely nothing to offer in helping solve the hailstone sequence. So just shut up. You're holding humanity back with your infinitesimal contribution.

## 1*3 +1 = 4 (0)

## Anonymous Coward | more than 3 years ago | (#36342934)

if it reaches 1 then the next would be 4 , so how could it possibly end in 1 ?

## Re:1*3 +1 = 4 (1)

## Surt (22457) | more than 3 years ago | (#36343090)

By definition, which the article submitter conveniently omitted.

## Re:1*3 +1 = 4 (1)

## Sique (173459) | more than 3 years ago | (#36344198)

Lets rephrase: It will always converge to the eternal sequence 4, 2, 1, 4, 2, 1, 4, 2, 1...

## Worst. Summary. Ever. (0)

## Anonymous Coward | more than 3 years ago | (#36342962)

Did a six year old write that? Is it really that hard to write a coherent sentence?

## Re:Worst. Summary. Ever. (-1)

## Anonymous Coward | more than 3 years ago | (#36343174)

You're completely pathetic.

By the way

HOSTS FILES for WELL ROUNDED SECURITY## LOL WUT? (0)

## Anonymous Coward | more than 3 years ago | (#36343104)

## Debunked already (4, Interesting)

## CSMoran (1577071) | more than 3 years ago | (#36343416)

## Re:Debunked already (1)

## assantisz (881107) | more than 3 years ago | (#36345798)

## Re:Debunked already (1)

## imsabbel (611519) | more than 3 years ago | (#36346244)

If you (metaphorically) go on the podium with your fly open and your dick hanging out, you deserve to be laughed off the stage.

I am waaay behind in my mathematics (since university), but even to me the paper looked rather fishy. If a problem has not been solved for decades, a prudent researcher would not push a paper like this out. And he surely wouldnt give interviews in the press talking how good he is and how he only recently started working on the problem...

## Re:Debunked already (0)

## Anonymous Coward | more than 3 years ago | (#36346398)

As a mathematician I do see this sort of back and forth exchanges, sometimes they are explicitly malicious, and other times they are more tactful. Mathematicians (myself included) often have a high standard of what they consider quality mathematics. You see debates over proof techniques, 'correct' hierarchy of ideas, etc. Very much like how programmers argue over stylistic issues, even when the results are the same (e.g. a program that does the same thing).

This has made some very critical thinking and meticulous people, if issues of presentation won't escape them, then certainly a flaw in an argument will not either. It is frustrating to hear public claims of proof of this or proof of that and have the presentation be not just flawed, but be so muddled that it takes considerable effort to discover that flaw (Remember the P = NP guy months ago? Same thing).

My mentor from when I was an undergraduate would put it this way: "If you can't teach it, you don't know it." Maybe an overstatement, but if you write a paper claiming something, then it should be so crystal clear that the possibility of error is unthinkable. You should be able to practically grok the argument on a first or second read. That's good mathematics. A proof is really you explaining why something is true. Just like in a college class you expect a lecture to make sense and you rate the quality of the professor on how well he explains something, you should also expect this from a proof.

## Re:Debunked already (2)

## ledow (319597) | more than 3 years ago | (#36348234)

Mathematics deals in certainties. An integer is odd (or prime, or divisble, or constant, or whatever), or it's not, by a very strict definition.

The problem stems from *proving* that. The mathematician (i.e. someone who studied mathematics at, say, degree level or higher) that can actually write a proof is surprisingly rare. The one that can write a perfect, undeniable, accepted, doubt-free proof in a single hit is would be seen as an absolute genius better than any other mathematician that's ever lived - more "clever" than Einstein, Hawking, etc. Because we're human and we always miss *something* in any proof ever offered for even the simplest of things.

Most proofs are like security programs on computers - they'll have holes with virtually certainty, and it's only accepted as "proof" when the holes have been found, poked at for 20+ years and every holes patched, countered and fixed. That's the process called "peer review". Look at any mathematical-based security process in computer science and you see the same pattern - cryptography, clever exploits, chroot-breaking etc. It takes decades to find MOST of the holes and an infinite amount of time to find all of them.

Any "proof" that comes from someone who hasn't been poking holes in others papers, or endlessly patching their proof for years based on the input of humongously talented third-parties is, history shows, 100% bullshit. Just read the story behind the solution to Fermat's Last Theorem, for example.

The problem is that the mathematical equivalent of "free energy" fanatics, etc., are people who claim to have proven something and yet have zero experience of providing actual proofs that are accepted by the mathematical community. It's elitest, sure, but it helps to filter out the nutters if you just ignore EVERYTHING posted on arxiv.org until it appears in a reputable, third-party-reviewed journal and has had ten rebuttals, all of which have been neatly countered using fabulously complex arguments.

You can now "prove" the four-colour theorem in an afternoon with a home computer and some mathematical knowledge. But it took *decades* to convince the mathematical community that a proof which relies almost entirely on the output of a computer to perform the grunt work was acceptable. And it was still in doubt from 1976 up until around 2005 as to whether it was actually a "proof" or not.

If someone claims to have solved a long-standing mathematical problem, with zero previous experience, treat them how you'd treat someone who claims that they can double the output of your car engine without using any extra energy source if you let them take the car apart. Yes, technically and historically, such things have been possible by certain marvellous innovations in the field. But would you really let him tinker with your car on the basis of that without knowledge of that person himself?

P=NP is a field that attracts people just like the "free/perpetual energy/motion" nutters, for instance. There's been at least three or four slashdot stories of "proofs" of that since the beginning of the year. None of them came to anything, all of them were firmly rebutted, and none of the authors were heard of again. It's not a general hostility in mathematics - just towards the time-wasters and nutters that think A-level maths qualifies them to provide a definitive proof of something new in an afternoon.

## Re:Debunked already (0)

## Anonymous Coward | more than 3 years ago | (#36348846)

Read the "about me" on that blog, he is a CS guy not a mathematician.

## Hailstone sequence (0)

## Anonymous Coward | more than 3 years ago | (#36343866)

Are these the hailstones that are the size of flat cylindrical objects (like coins) rather than the size of spherical objects (peas, golf balls' softballs ?

## Collatz Conjecture Explained... (2)

## dlgeek (1065796) | more than 3 years ago | (#36344350)

## 2^x (1)

## reasterling (1942300) | more than 3 years ago | (#36344526)

## Re:2^x (1)

## proverbialcow (177020) | more than 3 years ago | (#36344802)

The problem really becomes proving that for any {n: (n mod 4) = 2,3}, the sequence will eventually yield {n: (n mod 4) = 1}.

## 2^n (0)

## Anonymous Coward | more than 3 years ago | (#36346868)

Well, it seems you will always ultimately land on a number which is a power of 2. At that point, you'll be doing nothing but dividing by 2. You'll only be n steps from 1, with no escape.

The conclusion of the conjecture seems obvious. Showing that the sequence *must* at some point arrive at a power of 2, I'm sure, is quite hard.

## Can I try? (1)

## funnyguy (28876) | more than 3 years ago | (#36347212)

I'm not a math major, and I have very little experience with any proof. I've never written one. But I would first think this is a 3 part problem. One of n=even, n=odd and n=2^x. Proving all will become 2^x seems to be the goal.

Any even integer n divided by 2 is an integer half the value

Any odd integer n multiplied by 3 is an odd integer.

Any odd integer n plus one is an even integer

Through induction, any power of two in this sequence will end in 1. That is to say if n_0 is 2^x, then the sequence will always be even and always end in 1. 2/2 =1, 8/2/2/2=1.

As an infinite sequence, n will always become odd for any even n_0, not a power of 2.

Since odd n always have 1 added, they will eventually become a power of 2.

## Last step is flawed (1)

## Brannon (221550) | more than 3 years ago | (#36347254)

An odd n + 1 will always be even, but it isn't obvious that this will eventually lead to a power of 2.

## And of course (-1)

## Anonymous Coward | more than 3 years ago | (#36347334)

Here's a picture of the mathematician after his proof was re

butted [goatse.cx] .## Intriguing Observation (1)

## dtmos (447842) | more than 3 years ago | (#36348698)

Looking at the partial Hailstone tree [hubrisarts.com] as a network, it's interesting that the tree can fork only at every

othernode, even for nodes that aren't on the "power of 2" limb. There's no case in which forks occur in neighboring nodes.That would be a useful property to have in practical situations such as the generation of clock trees in digital integrated circuits. It would be interesting to see if similar rules could be applied to CAD systems.