Can this method be used to prove the Collatz Conjecture?

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Discussion Overview

The discussion revolves around the potential use of geometric patterns in graphs to approach the Collatz Conjecture. Participants explore whether visualizing the conjecture through graphical representations could yield new insights or methods for proving it, while also considering the implications of such an approach for mathematical problem-solving in general.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant suggests that the geometric patterns observed in a graph of total stopping time versus n could provide a new perspective on the Collatz Conjecture, potentially transforming it into a geometric problem.
  • Another participant acknowledges the existence of symmetries related to the Collatz conjecture but expresses skepticism about the utility of graphical representations beyond inspiration, emphasizing the challenge of finding a proof.
  • A third participant notes that while mathematical relations may correspond to aspects of the graph, proofs require unambiguous numerical representations rather than vague patterns.
  • A fourth participant references a related post, suggesting that there may be additional relevant discussions that could contribute to the topic.

Areas of Agreement / Disagreement

Participants exhibit a mix of agreement and disagreement. While there is acknowledgment of the potential usefulness of graphical representations, there is also skepticism about their effectiveness in leading to a proof of the conjecture. The discussion remains unresolved regarding the validity and applicability of the proposed geometric approach.

Contextual Notes

Participants highlight the need for clarity and unambiguity in mathematical proofs, indicating that the exploration of geometric patterns may not directly translate into formal proof methodologies. There is also an implicit recognition of the complexity and depth of the problem, suggesting that significant intellectual effort may be required to advance understanding.

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There is a graph showing n on its x-axis and its total stopping time on its y axis.

1140px-Collatz-stopping-time.svg.png

From here we can see that the points on the graph are not random at all; they have some kind of geometric pattern that is due to the 3x+1 in the odd case and x/2 in the even case. I have seen many attempts to prove the Collatz Conjecture but all that I have seen do not make any reference to the geometric patterns in the above graph. What if we work on establishing how the two cases' formulae relate to the above graph's geometric patterns and then extrapolate it? Might it be possible to find out something new from the formulae that would be really not apparent without this graph? Wouldn't it transform the original conjecture into a geometric problem, a new way of looking at it which may provide fresh new insights?

This method should work not for just the Collatz Conjecture,if I guess correctly. A math problem that could be graphed in some way would turn it into geometric problems which then could be solved/used to obtain new insights by geometry,wouldn't it?
 
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Symmetries of the Collatz conjecture are not unknown. Of course such graphics might be useful as an inspiration, but I think not for very much more. Assuming there is no counterexample (based on the many algorithmic hours on uncounted distributed computers), and if you will, also supported by the symmetries you mentioned, the real problem is to find a proof. Experience tells us, that such a proof can lead us far away from the original formulation of the problem, and it might need a genius like Andrew Wiles or Grigori Perelman and surely many years of research, to come up with a proof.

Since I don't know such a genius, I stick with Richard Kenneth Guy: "Don’t try to solve these problems!" American Mathematical Monthly 90, 1983, p. 35–41.
 
All the mathematical relations correspond to some aspects of this graph, but with numbers instead of "this pattern here, no not this, that one", because proofs need to be unambiguous.
 

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