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??


closed as primarily opinion-based by Yves Daoust, Professor Vector, user99914, I am Back, 5xum Nov 2 '17 at 8:48

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ Do you want one answer per mathematician ? $\endgroup$ – Yves Daoust Nov 1 '17 at 12:50
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    $\begingroup$ 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? $\endgroup$ – Fattie Nov 1 '17 at 12:52
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    $\begingroup$ 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? $\endgroup$ – littleO Nov 1 '17 at 13:07
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    $\begingroup$ 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: terrytao.wordpress.com/2012/09/18/… . In fact, Tao discusses why the Collatz conjecture is plausible here: terrytao.wordpress.com/2011/08/25/… $\endgroup$ – littleO Nov 1 '17 at 13:19
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    $\begingroup$ @Fattie I doubt that the Collatz-conjecture has been tested upto $10^{61}$. You probably mean $2^{61}$ $\endgroup$ – Peter Jul 5 '18 at 13:04

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.)

  • $\begingroup$ Astounding answer, thanks - ingesting ! $\endgroup$ – Fattie Nov 2 '17 at 21:19
  • $\begingroup$ 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. $\endgroup$ – Gottfried Helms Nov 27 '17 at 21:54
  • $\begingroup$ 2) 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) $\endgroup$ – Gottfried Helms Nov 27 '17 at 21:56

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