The 3 n + 1 conjecture is completely proved and it is very easy to do so.

Analysis:

Problem analyses are the first necessary, most important steps in the procedure of problem solving.

(3 n + 1) / 2 guarantee, (that's the value of good conjectures), that the repeat calculation values optimally aranged or evenly separated, that is, will not duplicate into loop or divergent before returning back into ever growing base. Here for its base of return, every step count.

3 n + 1 problem can go wrong only in the first few odd integers, its return trend more than 80% across the board guarantee it will return anywhere in just a few steps as long as it will not go into the loop. However not going into loop is guaranteed by the procedure hidden in the formulae across the board. That's why the larger the integer become, the the surer the process become.

Following are the facts based on this problem:

1. If first few integers tested are wrong, it will not affect the overall prediction of this problem.

2. All steps before the current integer tried become one (return point).

3. Current computer test (actual mathematical proof do not need this test) have let you have

   million and million step (different intergers) to try and you need only a few steps that will return

   you back, for any one integer.

4. The return points are initially built into the integer every other step (interleaved evenly).

5. The return factors are in the 2 exponent that make return trend is more than 80%.

6. There are countless root return points that reach destination in one 'step' in the whole integer

   range. These points with even r exponents (2 ^ r) are spread out into branches, if look through  

   graph and topology, it's like a web. However, they do not form loops and other integers

   randomly connect with the web.

7. The return points are in two grops, one grop is the all integers before current integer, the other

   group is the integers that were 'went through' by the integers before the current.

8. The tested region or integers grow evenly from beginning to end.

9. The return bases become larger and larger while the return process remain the same or stable.

All these are based on accurate calculation, not statistics.

The guarantee of always return can be proved strictly and mathematically (Make use of the fact that if you go forward one 'step', you will go backward 'four steps' at least, on average across the whole integer range, the fact that you can go 'endlessly' without repeating or going into loop if n become infinity, and the fact that the back points are built every other step, or every even step, so you walk without coming back is not possible).

The analyses are the informal proof process, by making sure the problem can be proved.

The Proof:

For the formal proof, there are different ways to do so and all proofs can be perfect.

I remembered I talked about this proof a few years ago in the network, they are not written down for publish as my many other proof works.

Anyway, its proof may use graph theory, topology, group theory, mathematical induction theory, and more. The important is to make sure that it is a real proof.

Now for double checking purpose, create a program to check all possible conditions consistantly, making a list of all possible data for analyses.

From the first reading this problem to finally completely proving it, I spent about 2 complete days to finish this work. Considering it is unsolved problem classified as very difficult, this time cost is fair. Anyway, it's not mine. It's belong to the nature.

By the way, the P and NP problem is also completely proved in any condition, and in doing so, some theorems to facilitate its prove are discovered. P and NP problem is more difficult than 3 n + 1 problem because the definitions of many concepts are not very well or even have errors, so correct proof of it is very hard.

Reading mathematical science and theoretical computer science is more difficult than reading bible, even for experienced profesionals, you will have this feeling for many years but if you persist, you will eventually get used to it.

Expected to improve.

But now, goodbye everyone.