Any odd number is squared and then one is taken away, any even number is divided by two, ill youll find it very interesting

I have also sent my proofs for Fermat’s Last theorem and Catalan’s Conjecture for publication. My proofs will shock the number theory world that such a simple proofs did exist for these problems.

All this work has been done in just last 2 months of time.

I left math in 1992 when I was fed up with University Politics and started working for Industries. I had never even imagined that I will solve these problems with such an ease. Proofs presented by me have already been checked by math professors, who have found these proofs quite fascinating. Now I am just waiting to have these published before you all will get a chance to see these proofs.

So just wait for few more months to see these proofs.

C: 1,5x+0,5 if odd; x/2 if even

Number 1 – stopping time: 0

Number 2 – stopping time: 1

Number 3 – stopping time: 5

Number 4 – stopping time: 2

= (0+1+2+5)/4 = 2

My general formula:

= 2/ln(0.75) * [n*(ln(1/n)+1)-1)]/n

The best results we obtain for large n, example n=1000:

= 39,89

My general formula:

= 41,08

[quote]Author’s note:
The reasoning on p. 11, that “The set of all vertices (2n; l) in all levels will
contain all even numbers 2n > 6 exactly once.” has turned out to be incomplete.
Thus, the statement “that the Collatz conjecture is true” has to be withdrawn,
at least temporarily.
June 17, 2011[/quote]

In the [3x+1,x/2]

We can look at this as two sets of data. Set one is the 3x+1 data and the other is the x/2 data.
Even and odd numbers are a part of both sets.
x=1 means 4,13,40,121…. it’s a progression
and given the constraints of dividing only numbers that have a power of two in them you can see that after a 3x+1 the even value that is present is a member of both sets but not a second even iif that even value can be divided by 4,8 and so on.
So the two systems share one element. The even after an 3x+10
I believe I was generating progressions to make the sets such as x=1 [2x] and x = 1 [3x+1] when I noticed the common element .
So they share a common value because in [3x+1] x can be an even value and even values are part of the progressions that make a complete set of values for [3x+1]
Remember I use brackets to denote cycle . I don’t know if anyone else does.

So the common element concept is another thing you guys might look at and see if it helps. I’m sure I have some C code some place . I work with C Code to explore things so I have a program for everything rather than a paper with equations.

Okay then.. If I can be of help..

Ernst_Berg@Sbcglobal.net

I too was fascinated by the 3x+1,x/2 and have written programs to use this.
My point of view was “it seems to work all the time so trust it.”
Anyway it’s been a dependable system.

I wanted to share what I know from exploring the 3x+1,x/2 form and I personally use notation of brackets [] to denote a cycle. so [3x+1,x/2]
For the 3 of 3x+1 we can say A so Ax+1,x/2 or Ax(+ or – ) y , x mod z
For all y that is a power of three there is one attractor for all input so 3x+3,x/2 will cycle 12,6,3.. and similar for all powers of three for Y of [Ax+/-y,xmodz]

Another thing that seems to be real is the idea of a cycle of values. The term Ring is already established as a proper math term so I will call it a ring as in a cycle.
We all know the 4,2,1,4,2,1 of 3x+1 so it is a cycle of three or a ring size of three. That is how I am using the term ring here.
Now since a system like 3x+5,x/2 has: Report for 3(x) + 5 Contains 6 Attractor(s) 1 19 5 23 187 347 I believe we are not looking at just 3x+1,x/2 but a larger family of systems and that family of systems has structure because it is obvious the attractors have relationships.

Proving 3x+1 means proving 3x+5 or y = 999999999999999…. for example.

Some attractors

Report for 3(x) + 1 Contains 1 Attractor(s)
1
———————————————-
Report for 3(x) + 3 Contains 1 Attractor(s)
3
———————————————-
Report for 3(x) + 5 Contains 6 Attractor(s)
1 19 5 23 187 347
———————————————-
Report for 3(x) + 7 Contains 2 Attractor(s)
5 7
———————————————-
Report for 3(x) + 9 Contains 1 Attractor(s)
9
———————————————-
Report for 3(x) + 11 Contains 3 Attractor(s)
1 13 11
———————————————-
Report for 3(x) + 13 Contains 10 Attractor(s)
1 13 131 211 259 227 287 251 283 319

Now I worked out one day that 3x+1 is one in a set of systems where 3x+1/x/2 was a part and it is based on the facts that if we express the process of any number in any [3x+y,x/2]we can write out the 3x part as one parity (“1″) and the x/2 as the other (“0″) in an even odd or binary string.
Then we can see that no two odd cycles ( if we represent 3x+y as a set or odd parity ) can be in a sequence sequentially because any 3x+y is followed by an x/2
so for the iterations of one cycle it is possible to have one of 2 outcomes 0 or 1.. In 2 iterations it is possible for 00,10,01 or three and in fact it is Fibonacci
2,3,5,8,13,21…

The set of all functions is then 2/1,3/2,4/3,5/5,6,8,7/13… I don’t know if “all functions” is appropriate I like to call it a Universe but I’m not qualified to name things like that outside of personal labels.
All of these can be explored. I looked into [7x-1,x/13] and adding or subtracting the Y can have different effects in different systems for different values.
I don’s see why more complex processing is not possible such as [3(x+7*12),x/2] or sum such thing that works.. I just typed that at random..
What they all have in common is pattern. Look at these from the parity expression they make. I don’t believe the point is the value of X it’s the pattern of value processed. In fact I consider the function to be the thing moving not value but I have several points of view I visit on these types of things..

3/2 shows us in the form [3x+/-y,x/2] that the patterns follow Fibonacci counts but 4/3 is different using 00,01,10 and 11 in it’s pattern. So too are 7/13 with long runs of one parity symbol before the other is seen and so on..
I figure there would be a practical use for generating patterns based on a single value for X. Proving them all is not on my todo list. I have explored a few other than 3/2 system.
Oh by the way if we want to generate a number that has over 1 billion iterations or more the “Reverse Collatz” I wrote can do that. It’s a crude program but was my attempt to explore what creates a value of a specific distance or stopping time.
I was looking for the natural language behind it all. The truth is the landscape changes when the Y changes. I conjecture that no two systems for Y where Y1 != Y2 have the same structure.
Alright.. I stand by what I wrote and offer this post as a contribution.
I have web sites and forums so if there is any need to have a forum I can add it to sites I already have up and running.
Currently I am working on the Million digit challenge and am finding the hard truth about data but I am not out of creative approaches yet.
all in all I like exploring these kinds of things and I assume they can be called dynamic systems. The result isn’t a number but a pattern from what I can tell.

Ernst
—-
Well that’s all good..

I also have a rudimentary program that can grow values that will be close to the natural iterations needed to reach 1 in a simple attempt at reverse [3x+1,x/2]

So that is what i did. I am open to discuss or share.

I was thinking that those cycles of value such as the 4,2,1 of [3x+1,x/2] may not be separate after all. Since non power of three Y values will have other than that Y in it’s attractor list is it possible that some systems connect in some way to other systems?

Well I saw structure and pattern in the attractors and there was an upward growth in the maximum value needed to express all the instances of attractor for a given system.

So I know that 3x+1 is not alone nor will explaining what it is doing only cover 3x+1

In fact I think we are just looking at two functions that share a common element in a struggle between two systems of 3x and x/2.
We are the force that causes it to cycle by artificially cycling it when 1 is a power of 2 and an odd. We allow all powers of 2 to be divided but one other wise it would head off into infinity as an endless series of x/2.

However, since this seems so stable I have envisioned using values to create pattern that can be used is an intelligent system perhaps in robotics where the functions of operations are defined by values but are visited by the current value of x in A(x)+/-Y,X/modZ] where A=3, Y=1 and Z = 2

Well That’s the short of it.. I’m available to share what I did so drop me a line if I can be of help ernst_Berg@Sbcglobal.net