Abstract
This paper studies the proof of Collatz conjecture for some set of sequence of odd numbers with infinite number of elements. These set generalized to the set which contains all positive odd integers. This extension assumed to be the proof of the full conjecture, using the concept of mathematical induction.
You can find the paper here: Preprint Collatz Theorem
(PDF) Collatz Theorem. Available from: https://www.researchgate.net/publication/330358533_Collatz_Theorem [accessed Dec 21 2020].
Dear Dagnachew Jenber
The first 11 theorems in your article provide a limit family of numbers that obey the Collatz conjecture and the number of steps needed to reach 1.
Unfortunately, in theorem (12), you have assumed that Collatz conjecture is true!!
In fact, your assumptions of the existence of b1 , b2 , ....bk-1 where k is finite
is exactly the Collatz conjecture, and the rest is an elementary computation of the number of steps to reach 1.
Can you prove that k is finite?
Obviously, if one assumes k is finite, then he assumes that Collatz conjecture is true.
Anyway, you have determined a nice family of numbers that satisfy the Collatz conjecture.
I wish you good luck to show k is finite.
Best regards
Dear Professor Issam Kaddoura,
Thank you very much for your time and consideration, Professor Issam Kaddoura. This is a valuable comment and it is what I want. In fact, in Theorem 8, I proved that the existence of natural number n=2k-1, for all k belongs to a natural number such that Collatz Conjecture is true and in Theorem 10 the number of steps needed to reach 1 for 2k-1 if the number of steps needed for 3k-1 is known, that is they are related by $N_{2k-1}=2+N_{3k-1}, N_{1}=3$. As you see this is a recurrence relation to determine the number of Collatz function operations needed for each natural number to reach 1 recursively. Moreover, you can observe that $2k-1$ and $3k-1$ can not be equal, which means the number of steps for each number is uniquely determined from the other and the existence can be seen as well. Obviously, $n=2k-1, \forall k\in \mathbb{N}$ Theorem 8 tells us that finite $n$ implies the finiteness of $k$. In Theorem 12, as you stated in your comment, I assumed that Collatz conjecture is true, that is why I thought I proved it in the previous theorems. For the existence of $b_{1}, b_{2}, \cdots, b_{k-1}$ please see Theorem 9. This is what I have seen, If I miss your comment please tell me. Moreover, if there are certain things that needed to be added that you see, not seen for me, please let me know. Thank you very much for your time and valuable comments. Have a blessed day.
Best regards,
Dagnachew J.
- Badges
- Science topic
- Similar topics
- Mathematics