The Unrepeatable Function

The unrepeatable function is

NRF (P) =

{ 3 ^ (a - 1) + 3 ^ (a - 2) x 2 ^ r (1)  + 3 ^ (a - 3) x 2 ^ r (1) x 2 ^ r (2) + .....

+ [ 3 ^ (a - i) x 2 ^ r (1) ..... 2 ^ r (j) ] + ..... + [ 3 ^ 1 x 2 ^ r (1) ..... 2 ^ r (a - 2) ] + [ 2 ^ r (1) ..... 2 ^ r (a - 1) ] }

/

{ [ 2 ^ r (1) x..... x 2 ^ r(j) x ..... x 2 ^ r (a) ] - 3 ^ a }

where a is an integer for the munber of 3 X + 1 advance operations,

r(j) is an integer function for the munber of returns of division by 2

untill another odd munber is reached, and NRF (P) will return P if a loop

is found. As a matter of fact, NRF (P) will always be less than one.

Throughtout the whole odd integer range, for any starting odd munber,

NRF can not find any repeat number. The only exception is when

starting odd munber = 1, the destination.

The Odd Number Function

The odd number function is

ONF(P) =

{ (3 ^ a) x P + 3 ^ (a - 1) + 3 ^ (a - 2) x 2 ^ r (1)  + 3 ^ (a - 3) x 2 ^ r (1) x 2 ^ r (2) + .....

+ [ 3 ^ (a - i) x 2 ^ r (1) ..... 2 ^ r (j) ] + ..... + [ 3 ^ 1 x 2 ^ r (1) ..... 2 ^ r (a - 2) ] + [ 2 ^ r (1) ..... 2 ^ r (a - 1) ] }

/

[ 2 ^ r (1) x..... x 2 ^ r(j) x ..... x 2 ^ r (a) ]

where P is a starting odd munber, and ONF(P) return odd number for step a.

ONF(P) will have P as its average maximum fluctuation point because

the minimum average return tendency roughly = (2 + 4) / 2 = 3, the same as 3 X + 1 advance rate.

Acceleration Networks

There are many acceleration networks on the whole integer range,

to name a few, they are:

1. The stop points network: 2 ^ e, where e = 1, 2, 3, ......, which satisfy immediately;

2. The para-stop points network: 3 x 2 ^ e, where e = 1, 2, 3, ......, which return 3;

3. The para-stop points network: 5 x 2 ^ e, where e = 1, 2, 3, ......, which return 5;

4. The para-stop points network: 7 x 2 ^ e, where e = 1, 2, 3, ......, which return 7;

.....

The list is endless.

To formally prove, you need to use random sample statistic theory for the whole integer range

using the following facts:

1. The even integers are interleaved by return factors 2 and 2 x 2 ^ e exactly one after another,

    where e = 1, 2, 3, ......

2. The odd nuumbers advances by 3 X + 1 follow the above orders specified.

Plus the above two functions.

Start from any initial number P0, followed by P1, P2, P3, etc. until reaching least number

as there is no loop,meaning: no matter how long you travel, you will not repeat any P

and eventually, you will step on the large return grave.