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I noticed Wikipedia says that

most mathematicians who have looked into the problem think the conjecture is true because experimental evidence and heuristic arguments support it

(seen on Wikipedia, late 2017).

  1. Is this just wrong? i.e. it's simply not true that "most" mathematicians who have looked into Collatz think Collatz is true - ?

  2. My question then, if they do tend to think it is true: why, very specifically, do they think that?

The fact that it's been tested up to about 10^61 doesn't mean much, many such theorems turn out to be wrong in such cases.

The "probabilistic" "3/4" observation is not helpful (we already know it's "probably" true from any cursory examination. So what?)

As I understand it Krasikov and Lagarias showed that (in a word) "most" numbers definitely go to one, but we already know that.

(The work of Kurtz and Simon is beyond me but I believe it comes closer to showing - if anything - that the problem is "proven unsolvable", but that if anything would seem to dismiss the idea that "most mathematicians think it is true".)

Is there anything else?

What's the deal on all this? Has there been any further recent breakthroughs (beyond my laughable knowledge level!) which would suddenly mean "most mathematicians who have looked into Collatz think" it is true?

Again, my question is, and thanks in advance, why very specifically do these folks think that??

Fattie
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    Do you want one answer per mathematician ? –  Nov 01 '17 at 12:50
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    hi Yves, that's extremely silly. With other similar situations, one can give specific, actual reasons why "we tend to think the conjecture is true, even if there is no proof yet". What, then, are these reasons in the Collatz case? – Fattie Nov 01 '17 at 12:52
  • While it's not entirely a valid parallel, I think it's worth mentioning Russel's teapot. So far we have not seen any evidence of numbers that do not go to $1$, and while the idea that such numbers exist is less ridiculous than the teapot in orbit around the sun, mathematicians are swayed by empirical evidence such as this. Note however, that there is a big gap between believing something might be true, and being convinced it's true. I think "most mathematicians" are of the former type, while you seem to place them closer to the latter. – Arthur Nov 01 '17 at 12:55
  • hmm, @Arthur, the only problem with that is: it seems to me the one and only, even hint we have that it might be true, is that, we've tested it using a pocket calculator up to 10^61. (Am I right?) But that means simply nothing, it's the very definition of "a mathematical puzzle where all we know is that we've tested it a bit" - it's just literally begging the question. Even something as lightweight as wikip instantly gives various examples like the Mertens conjecture, where testing meant nothing. – Fattie Nov 01 '17 at 13:01
  • Though I’m not a mathematician yet at this moment, I would say, I hope that it’s true——It’d be fantastic. – BAI Nov 01 '17 at 13:06
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    I think it's a very good question -- what are the heuristic arguments that lead some mathematicians to believe that the Collatz conjecture is likely true? – littleO Nov 01 '17 at 13:07
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    As I said at the end of my comment, I believe that the most common attitude towards this problem is "Believing it might be true" (which I just see as "have no reason to believe it untrue"). You seem awefully fired up about something so entierly inconsequential as the personal beliefs of mathematicians. Is it really such a bad thing that we have some beliefs that aren't rooted in firm evidence? Keep in mind that in mathematics, more than any other scientific field, we are capable of keeping opinion and fact separated, so carrying preconceived notions aren't as damaging as in other fields. – Arthur Nov 01 '17 at 13:09
  • That being said, I would also be interested to see heuristics other than "we tested a lot of numbers". I just think you're getting a bit too worked up about it. (PS. when I say "keeping opinion and fact separated" I am aware that things like conventions, such as whether $0$ is a natural number, is a gray area between them that some mean is a fact, and some mean is an opinion. That's not what I'm talking about. I'm talking about things that are clearly one or the other) – Arthur Nov 01 '17 at 13:10
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    The style of the question might be antagonizing people a bit, but it's often useful to understand heuristic arguments in favor of a conjecture. For example, Terence Tao discusses the probabilistic heuristic justification of the ABC conjecture here: https://terrytao.wordpress.com/2012/09/18/the-probabilistic-heuristic-justification-of-the-abc-conjecture/ . In fact, Tao discusses why the Collatz conjecture is plausible here: https://terrytao.wordpress.com/2011/08/25/the-collatz-conjecture-littlewood-offord-theory-and-powers-of-2-and-3/ – littleO Nov 01 '17 at 13:19
  • I like the tone of the question a lot more now. Thank you. – Arthur Nov 01 '17 at 14:57
  • I am not sure that mathematician who worked on Collatz are just "believing" because it was tested up to some magical number. I won't talk for them, but I myself am convinced the conjecture is true because of the structure behind it (which is more visible when you look at the Collatz tree, coverage, and the way they are all related). If it was only tested up to 10 thousand would not have any effect on my "conviction". After all, it took 300 pages to prove 1+1=2, the fact that something is hard to prove does not always mesure the degree of conviction you might have in something. – Collag3n Nov 01 '17 at 17:04
  • I can't answer as your good question is on hold. Unless you want to lose a lot of rep, it's best not to discuss Collatz on here. However; having studied the problem in depth the strongest argument for its truth in my opinion is that unlike most similar questions, it is not possible for there to be some isolated counterexample. The existence of any counterexample would mean the existence of at least one tree of infinitely many counterexamples, and this would require the structure of the known tree to suddenly change and become less dense in the integers above a certain range. – Robert Frost Nov 02 '17 at 18:47
  • There are many other pointers; such as the close relationship to a number of other very difficult unsolved problems relating to metrizing the distances between powers of 2 and powers of 3. One of the best is Mahler's 3/2 problem. Another (proven) is Catalan's conjecture. It is likely that these are related to the nontrivial kernel of the 2-adic numbers and the complex relationship this generates between multiples of $2$ and multiples of $2+1$. The problem looks to be a fractal closely related to Sierpinski's triangle when superimposed over Pascal's triangle. As such it seems to encapsulate... – Robert Frost Nov 02 '17 at 18:52
  • ... a deep and important component of number theory. The most intriguing possibility is that it lies at the heart of the Riemann hypothesis via the identity $\sum_n \dfrac{3^n-1}{4^n}\zeta(n+1)=\pi$ although at the present time I am the only person I know of investigating this. – Robert Frost Nov 02 '17 at 18:57
  • hi @Collag3n, that is amazingly insightful (thanks), I'm surprised you wouldn't make an answer ........ – Fattie Nov 02 '17 at 21:20
  • Would it be easier to share our thoughts here? https://chat.stackexchange.com/rooms/68496/everything-collatz-conjecture – Griffon Theorist697 Nov 11 '17 at 01:13
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    @Fattie I doubt that the Collatz-conjecture has been tested upto $10^{61}$. You probably mean $2^{61}$ – Peter Jul 05 '18 at 13:04
  • @Collag3n "It took $300$ pages to prove $1+1=2$" ? Do you have any reference for this claim ? – Peter Jul 05 '18 at 13:07
  • @RobertFrost The argument that no isolated counterexample can exist does not convince me. All the counterexamples would be very large. That sequences can change their behaviour dramatically very very late, can be seen by the Goodstein-sequences. I see no reason why the situation in the case of Collatz's conjecture could not be similar. – Peter Jul 05 '18 at 13:10
  • @Peter I take that on board. I, like you, am not certain - i.e. to the point I would totally rule out the existence of counterexamples. But having studied it in some depth, I would put the probability it's true arbitrarily close to $1$. Aspects of the problem repeatedly suggest convergence is a mathematical identity rather than a very likely event. I have no doubt if I were better at maths I could complete it now, as I have the flavour of the proof... – Robert Frost Jul 05 '18 at 14:29
  • ...But I don't have the maths ability yet to quite get over the finishing line. If a good mathematician were to have a sensible conversation with me about it, I think we could get it done. – Robert Frost Jul 05 '18 at 14:29
  • @Peter if I were to make my simplest, strongest argument that the conjecture is true I would point out that the Collatz graph $x\mapsto 3\frac x4+2^{\nu_2(x/4)}$ is self-similar in $\Bbb Z[\frac12]/\Bbb Z$ and enjoys the same self-similarity $x\cong \frac x2$ that the group itself enjoys if you were to set odd numerators equivalent. – Robert Frost Jul 05 '18 at 14:47
  • Let us continue this discussion in chat. – Peter Jul 05 '18 at 14:59

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I cannot stand for all mathematicians, however I can describe in detail why after roughly a year of research into the Collatz Conjecture why I personally believe the Conjecture could be true and why the supporting evidence $87*2^{60}$ and the $“3/4”$ argument are significant.

The belief that the Collatz Conjecture always reaches one may stem from mathematicians and others connecting this evidence to what they understand about the Conjecture. Most people who were either introduced to the Collatz Conjecture or heard about it and became curious at some point worked out some of the Collatz sequences by hand or with code. In doing so, they get the sense of why there’s so much confusion and gain a first-hand experience of the “randomness” generated by the algorithm.

When looking at the $87*2^{60}$ evidence or the $“3/4”$ argument they may conclude, “This supports what I worked out on paper or on my computer already, so this evidence must make some sense and therefore the Collatz Conjecture could be true.” If they choose to spend more time on the Conjecture, this idea may be reinforced over time. This could also lead to the opposite idea where some people believe there must be some gigantic number out there that disproves the Conjecture. Personally, every time I worked on the problem, I became more and more convinced that the Conjecture is true, but it needs to be proven and the algorithm needs to be dissected and explained.

Of course, this only explains where the initial perspective of my own and possibly others came from, and one reason some people may have for believing this evidence has some (or no) meaning. However, interpretation is not the only reason this evidence may mean more than just that.

Without context, any statistic, ratio, or really big number does not mean or suggest anything, and just because a big number or well received ratio was derived from the problem it came from does not mean the context of the original problem supports it. This may not the case for these two pieces of evidence.

The $87*2^{60}$ evidence and $“3/4”$ make more sense as evidence when modified Collatz rules are considered where if a number $x$ is odd, multiply by $ax+b$, and if $x$ is even, perform $x/2$. While tweaking with these rules can sometimes lead to drastically different results, these are the closest rules that can be referenced as additional context since these algorithms share some basic fundamental rules despite not being simpler generalizations most of the time. As a result, we can look at rules that would have ‘false conditions’ such as another cycle or wandering off to infinity.

Modified Collatz rules such as $3x+5$, $3x+7$, and $3x+11$ have multiple loops for $x>0$. What is interesting here is these loops are fairly accessible; Most of these loops can easily be found by hand. As far as we know, none of these rules have strange loops that start in the millions, or trillions, or anything like that. Another rule $3x+3$ also seems to share the same loop behavior as $3x+1$ but instead of going to one it goes to 3. After seeing some other examples of Collatz loops, it begs the following question: If such a loop existed for $3x+1$ among the googleplexquadrillionsmillions or whatever, why does it exist and why have we come across more examples of loops with smaller numbers in the modified Collatz rules?

A similar approach can be applied to the $“3/4”$ evidence. The modified rule $5x+1$ seems to have a bizarre [1-6-3-16-8-4-2-1] loop and then once you start with 7, the numbers seem to explode towards infinity, occasionally shrinking every-so often along the way. This speculation makes more sense when the formula $log(5)-2log(2)$ expresses a value larger than one, supporting the observed “infinite” behavior. Having this as something to compare to, the $”3/4”$ argument now makes much more sense as a possible explanation for why the Collatz Conjecture on average does not explode where $5x+1$ seems to do so.

I know it may seem like I am cheating, a modified Collatz rule is certainly not the same as the original $3x+1$ problem. However, at least for the $3x+b$ rules, I believe there may be a relevant connection aside from convenience. For instance, the Collatz Conjecture seems to be embedded into some of the positive integers of rules where $b$ is odd and $b>1$. For example, apply $3x+5$ for $x$ = 65. The resulting trajectory will be a multiple of the trajectory for 13 iterated by the algorithm $3x+1$. Therefore, I assume it may be possible there is at least some relation between the Collatz Conjecture and these modified rules.

[$3x+5$] 65->200->100->50->25->80->40->20->10->5->…

[$3x+1$] 13-> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2-> 1->…

Obviously, all of this is speculation until a formal and correct proof comes out or provable explanations for these patterns emerge. I hope this at least gives the impression of where I stand on this issue without sounding too much like a crank, and makes these pieces of evidence seem more appropriate when representing a possible case for what the proof of the Conjecture might or might not look like.

(note: Sorry for the long response. I did my best to be specific.)

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Griffon Theorist697
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  • Astounding answer, thanks - ingesting ! – Fattie Nov 02 '17 at 21:19
  • Two additions. 1) for the $3x+r$-case. If we allow fractional numbers then in the $3x+1$-problem we can have cycles on such fractional numbers. Let the smallest one be $a_1=p/q$ with $p,q$ different primes. Then with $r=q$ the $3x+r$-case has a cycle at $a_r=a_1 \cdot r$ which is then an integer. – Gottfried Helms Nov 27 '17 at 21:54
  • The idea, that if there is no cycle in small integers then perhaps in the zentillions... The Collatz-cycle problem has the nice property, that there is an upper bound for the members of a cycle - this is (a bit "fuzzy") depending on the cycle-length $N$, so it implies a conceptual tendency *against* an argument of big numbers. That two conditions might make it again easier for some mathematician to trust that the conjecture is true after up to $a_1 < 87 \cdot 2^{60}$ there's no cycle. (The question of divergence seem to less frequent been considered beyond the statistical reason)
    • – Gottfried Helms Nov 27 '17 at 21:56