0

Is there a known so specific mathematical reason that makes it difficult to solve the Collatz Conjecture?

Clearer: What is the specific mathematical reason behind the origin of the Collatz Conjecture that makes it difficult to solve it?

I'm looking for a specific reason known to mathematicians.

Thank you.

Learner
  • 745
  • It is not clear that it is impossible to solve the Collatz Conjecture. It is just difficult, like Fermat, which was really difficult, although it looked not so hard perhaps. – Dietrich Burde Mar 30 '19 at 14:35
  • @DietrichBurde I fixed thank you. – Learner Mar 30 '19 at 14:36
  • Possible duplicate of Why is the Collatz Conjecture so difficult to prove or disprove?. – Dietrich Burde Mar 30 '19 at 14:39
  • @DietrichBurde I'm looking for a specific reason. – Learner Mar 30 '19 at 14:42
  • 3
    Impossible is not a word to throw in lightly. I would say extremely hard and one reason is its deep connection with transcendence theory, specifically that even a weaker form of collatz, namely that the only cycle is $1,2,4$ implies a deep result about separation of powers of $2, 3$, namely that $2^a-3^b > 0, a,b$ positive integers, implies $2^a-3^b >> (1+c)^b, c>0$, small absolute constant and while stronger results are known they all use deep methods of transcendence theory and no other methods are known to get such separation between powers of $2,3$. – Conrad Mar 30 '19 at 14:49
  • @Conrad thank you for comment. – Learner Mar 30 '19 at 14:58
  • 1
    For more detail check terry tao's blog article on the subject - easy Google search as am on phone and hard to include link – Conrad Mar 30 '19 at 15:01
  • 1
    The question may not be good, but why downvote? – Learner Mar 30 '19 at 20:36
  • "...let us remark first that the true level of difficulty of any problem can only be determined when (and if) it is solved. Thus there can be no definitive answer regarding its difficulty. The track record on the 3x + 1 problem so far suggests that this is an extraordinarily difficult problem, completely out of reach of present day mathematics. Here we will only say that part of the difficulty appears to reside in an inability to analyze the pseudorandom nature of successive iterates of T(x), which could conceivably encode very difficult computational problems...." Chap 1, pg 4 (...cntd...) – Gottfried Helms Mar 30 '19 at 23:50
  • (...cntd...) From the Lagarias book "the ultimate challenge". Note there are already many answers for this question here in MSE around. – Gottfried Helms Mar 30 '19 at 23:52

1 Answers1

4

Proving the Collatz conjecture is equivalent to finding inconsistent constraints on the least counterexample $n$.

On the one hand, $n$ must be odd, say $2k+1$; on the other hand, if $3k+2$ were even we'd again have a contradiction, so $k$ is odd i.e. $n=2k+1=4l+3$ (say). Similarly, $\frac{3n+1}{2}=6l+5$ gets mapped to $9l+8$, which can't be divisible by $4$, so nor is $l$.

And we can go on like this constraining $n$ modulo powers of $2$, but it doesn't seem we can get the number of "legal" residue classes to go down from $1$ to $0$; indeed, it looks like we now have three options modulo $16$. Sure, we can probably constrain that further, but the problem doesn't seem to want to achieve a $o(m)$ upper bound on the legal number of modulo-$2^m$ residues classes. Managing something that's $o(2^m)$ just isn't enough.

It's not for want of trying. The Ultimate Challenge: The 3x+1 Problem (Lagarias 2010) is a book-length summary of where we're at so far, and why where we are doesn't look all that promising. You really should read that, in addition to any answers we manage here. He devotes chapters to such approaches as dynamical systems, Markov chains and ergodic theory, and stochastic models. My copy isn't searchable; I only wish I could find this quotation of it in context, to know why he says it:

this is an extraordinarily difficult problem, completely out of reach of present day mathematics.

J.G.
  • 115,835
  • 1
    (+) thank you for answer. – Learner Mar 30 '19 at 15:29
  • "...let us remark first that the true level of difficulty of any problem can only be determined when (and if) it is solved. Thus there can be no definitive answer regarding its difficulty. The track record on the 3x + 1 problem so far suggests that this is an extraordinarily difficult problem, completely out of reach of present day mathematics. Here we will only say that part of the difficulty appears to reside in an inability to analyze the pseudorandom nature of successive iterates of T(x), which could conceivably encode very difficult computational problems...." Chap 1, pg 4, pdf-version – Gottfried Helms Mar 30 '19 at 23:39