Dear @Baris, I suggest that you write more details about the Collatz Conjecture.

Best wishes, Abedallah Rababah

Dear Baris:

  What kind of collaboration are you looking for? Are you willing to sign a collaboration document?

  Thank you in advance and best regards,

         Phd, Andrés García. UTN-FRBB, Argentina

Dear Abedallah and Andrés,

Thank you for responding to my question. This is something I took interest in, and is totally separate from my normal area of research (cardiovascular sciences/medical physics).

Essentially, I carried out an investigation of Collatz series and tried to find out why they always seem to converge to unity even though they can initially seem to have divergent behavior (e..g. in the case of the initial starting number 27).

It seems clear from the literature that analytically it may be difficult to prove/disprove the conjecture but I have found some empirical results which may help explain why they do converge and by collaborating with a professional mathematician(s) I thought we could convert the very primitive paper I produced into something of a more proper manuscript for publication somewhere.

Please kindly let me know if you would like me to send you a copy of the primitive paper I prepared so far. It essentially highlights some sort of a symmetrical distribution (bit like a Gaussian) appearing in Collatz series; perhaps we can substantiate it mathematically as well (with your expertise).

Many thanks &

Kind regards,

Baris.

p.s. I do not mind signing a collaboration document if necessary.

Dear Baris:

  Thank you for your answer, It sounds very interesting and by the way I was working with that conjecture some time ago. Could you please share with me that paper you mentioned?.

  The document I aforementioned is because in our university, it is important to have documents establishing the collaboration with other collegues.

  Keep in touch and best wishes,

        Andrés

PD: My e-mail is [email protected]

Hi Andrés,

Please do forward me the collaboration agreement if you think it is necessary. For me, this is something I am doing in my spare time, so it is not related to the work I do at my university.

I am forwarding the preliminary paper to your email address. It is rather primitive at the moment, but I think if we can provide some supporting mathematics, it may be interesting!

Kind regards,

Baris.

Dear  Dr. Abedallah M Rababah 

I have seen Collatz conjecture.   I can present  my   probabilistic approach:  Let $x$ denotes the random natural number. Let denote by $P(x)$ the probability that Collatz sequence starting at $x$ will return at 1.

If $x$ is divided to $2$ the $x$ is sending to $x/2$ with probability $1/2$ and now  the probability that Collatz sequence starting at $x/2$ will return at 1 will be equal to $P(x/2)$

or

If $x$ is not divided to $2$ then  $x$ is sending to $3x+1$ with probability $1/2$ and now the probability that Collatz sequence starting at $3x+1$ will return at 1 will be equal to $P(3x+1)$. Using total probability formula we get the following 

functional equation:

$P(x)=1/2P(x/2)+1/2P(3x+1)$.

Notice that  each  constant  function satisfies  this  equation, i.e. $P(x)=c $ for each natural number $x$.   Taking into account that   $P(2)=1$ we deduce $c=1$. 

Hence we have  proved that  the probability that Collatz sequence starting at an arbitrary natural number will return to 1 is equal to 1. 

P.S. Since for an arbitrary  natural number  $n$ there exists  only one Collatz sequence starting at $n$  and  $P(n)=1$ , does this means that  this is proof  of the  Collatz Conjecture?

Dear Gogi,

What do the $ signs signify?

I am not a mathematician but to me it seems like P(x)=1/2P(x/2)+1/2P(3x+1) would not work as P(x/2) will be undefined if x is odd (as x/2 will not be a natural number) and P(x) will be undefined+1/2 or undefined+0.

I think more correctly, it would be P(x)=P(x/2) if x is even and P(x)=P(3x+1) if x is odd but I doubt it is of any help as this would be the case anyway and P is really a binary flag (i.e. 1 or 0) not really a probability measure?

Kind regards,

Baris.

Dear Baris Kanber

Hi

You are right.  Denoting by $P(n)$ a probability that Collatz sequence starting at $n$ will return at 1, we know that $P(n)=1$ or $P(n)=0$ for each $n=1,2\cdots$.

We have that $P(n)=P(2n), P(2n+1)=P(6n+4)$ for $n=1,2,\cdots$. It is checked that for sufficient large natural numbers $P(n)=1$. It is obvious that a function $P_1(n)=1$ for each $n=1,2, \cdots$ satisfies equations above and at the present time agree we results of computations, but we do not know whether equality  $P_1=P$ ( Collatz  conjecture) holds true.

P.S. Sorry  for my  first  answer . It  was not correct and  needs  some mathematical formalizations . 

Best regards,

Gogi

Dear Baris

I have submitted manuscript   entitled by  "A behaviour of relative frequencies of even and odd numbers in the sequence of $n$-th coordinates of all Collatz sequences" at 

https://arxiv.org/submit/1115560/view

Best regards,

Gogi

Hi Gogi,

Thank you for sending me the link, but I can not access it. Have you got the Pdf at all?

Hi  Baris

see

https://www.researchgate.net/profile/Gogi_Pantsulaia/publications

Best regards,

gogi