Analyzing the Collatz Conjecture Using the Mathematical Complete Induction Method
Round 1
Reviewer 1 Report
Your proof of the Collatz conjecture in section 3 which you used the mathematical complete induction method is true and straightforward. Is there a generalized Collatz conjecture? Or is this the only possible one which the last terms are the cycle 4, 2, 1? What about the other possible cycles?
Author Response
Dear referees,
We are very grateful for reviewing our paper and writing to us your reports with your opinion and suggestions.
Here you can find bellow, the list of changes of our manuscript, following your suggestions:
1.- In Introduction: we remark that in our paper we are showing a proof of the Collatz conjecture as it was stated for positive integers. We haved added:
- A) Before of the paragraph where it says “In this paper, we demonstrate…”, we add:
“There are extensions in other domains, such as iterating through all integers, obtaining in this case other cycles than 4, 2, 1. However, it is not our aim to study these extensions in other domains. Although, quite possibly, the method of demonstration we show could be used to prove the generalised conjecture.”
- B) In the paragraph where we say “In this paper, we demonstrate…”, we add:
“In this paper, we demonstrate the Collatz conjecture as it was stated for positive integers number using the mathematical complete induction method.”
2.- In page 5, X_{S+u} the S+u is all written as a subindex.
3.- All that is explained in page 6, we consider that it is absolutely necessary to understand why this limite is equal to 1. Therefore, we can not remove/change anything.
4.- We add the MSC code after keywords.
With our very best regards,
Dr. Christophe Jouis and Prof. Mercedes Orús-Lacort
Reviewer 2 Report
The conjecture of Collatz is a apparently simple problem of mathematics, but old and unsolved. More exactly, this conjecture is a one of the biggest from the number theory research field of interest. Let us consider an arbitrary positive integer number n. If this number is even, then it will be transformed in the half, n/2. If the number n is odd, then it will be transformed in the number 3n+1. The Collatz's conjecture say that starting with any positive integer n and after applying of succesive previous transformations, always reach 1. The single the loop of this tranformation is 4,2,1.
Despite recent progress in the solving of this conjecture, we still don't know if there exist a positive integer number who can escape its infinite loop which is formed with 4, 2 and 1. Authors of the present manuscript presented a very nice tentative of proof for the conjecture of Collatz, using the mathematical complete induction method. Until now the conjecture was confirmed by computation until the huge number $2^{68}$, that means 295,147,905,179,352,825,856. This number is a very, very huge, almost 300 millions of trillions.
In line 176 of the manuscript (page 5) there is a little mistake at "it means that the limit of xS+u ... ". Surely, you may put $x_{S+u}$.
From my point of view, because the authors use the limit when the number of iterations tend to infinity (see R and k tend to infinity), then I think that this can be a big problem! If R and k tend to infinity, then the number m (and also m+1) may be go to infinity. In any case, for m (and for any number less than m) was supposed that there are a finite number of successive iterations when reaches 1.
Here it's a contradiction with the mathematical complete induction method. See the pages 5 and 6.
Sorry, but on page 6 I got lost in the details of the discussion with a, t, w, ...
Here maybe it would be better to simplify the presentation of the proof in this point.
May be is better to not claim that this is a really proof of this famous conjecture of Collatz! Because I'm not at all sure that this proof is correct!
It would be nice to be true!
I don't know, maybe "mathematics is not yet ripe enough for such questions" (by Paul Erdos)
Author Response
Dear referees,
We are very grateful for reviewing our paper and writing to us your reports with your opinion and suggestions.
Here you can find bellow, the list of changes of our manuscript, following your suggestions:
1.- In Introduction: we remark that in our paper we are showing a proof of the Collatz conjecture as it was stated for positive integers. We haved added:
- A) Before of the paragraph where it says “In this paper, we demonstrate…”, we add:
“There are extensions in other domains, such as iterating through all integers, obtaining in this case other cycles than 4, 2, 1. However, it is not our aim to study these extensions in other domains. Although, quite possibly, the method of demonstration we show could be used to prove the generalised conjecture.”
- B) In the paragraph where we say “In this paper, we demonstrate…”, we add:
“In this paper, we demonstrate the Collatz conjecture as it was stated for positive integers number using the mathematical complete induction method.”
2.- In page 5, X_{S+u} the S+u is all written as a subindex.
3.- All that is explained in page 6, we consider that it is absolutely necessary to understand why this limite is equal to 1. Therefore, we can not remove/change anything.
4.- We add the MSC code after keywords.
With our very best regards,
Dr. Christophe Jouis and Prof. Mercedes Orús-Lacort
