Collatz-Intro - Resources: Some Tables

Workshop Recreational Mathematics

Gottfried Helms Univ. Kassel mailto: helms at uni-kassel www: math-homepage

Resources: Some Tables

An observation of a property of the transformation-rule respective their exponent A gave reason to classify all odd numbers (as candidates for the loop members) in the following scheme, depending on their transformation-exponent.

Consider the following table:

Class name	Class	Reduction-exponent A	Reduction-power of 2	Result of one transformation
C1	a = 4i+3	1	2	a' = 6i+5
C2	a = 8i+1	2	4	a' = 6i+1
C3	a = 16i+13	3	8	a' = 6i+5
C4	a = 32i+5	4	16	a' = 6i+1
C5	a = 64i+53	5	32	a' = 6i+5
C6	a = 128i+21	6	64	a' = 6i+1
C7	a = 256i+213	7	128	a' = 6i+5
...	...	...	...	...

Each class has infinitely many members, depneding on the free parameter i which is allowed to vary over all nonnegative integers. Any class of numbers requires the same power-of-2 in a regular collatz-transformation.

It is easy to prove, that the union of all these classes build exactly the set of odd natural numbers. This is important, since we can safely assume, that all candidates for an eventual loop are in one of these classes.

The additional benefit of this classification is, that in any formula, which involves the exponents A,B,C of the reducing transformation (division by 2^A) I implicitely know, which class of numbers the related member of the loop belongs to. This is a crucial requisite for the proof below.

For instance all the following numbers : 3,7,11,15,... require a transformation with A=1 which means 2^1 = 2:

a = 4*i + 3;

i=0 a= 3 a'=T( 3;1) = 3* 3+1 / 2^A = 10/2 = 5 means (a->a') = ( 3->5)
i=1 a= 7 a'=T( 7;1) = 3* 7+1 / 2^A = 22/2 =11 means (a->a') = ( 7->11)
i=2 a= 11 a'=T(11;1) = 3*11+1 / 2^A = 34/2 =17 means (a->a') = (11->17)

and so on.

All numbers of this form C1 can only be divided by powers of 2 with exponent 1, so if I discuss a transformation

a' = T(a;1)

this requires, that a *must* have the form of 4*i+3.

Also the structure of the result is obvious; and that structure can be uniquely written as depending on i; in any case it is just

a' = 6i + 5

i=0 a = 3 -> a' = 6*i + 5 = 5
i=1 a = 7 -> a' = 6*i + 5 = 11
i=2 a = 11 -> a' = 6*i + 5 = 17

and so on.

For instance, this makes the disproof of the 1-step loop with exponent A=1 very simple:

a' = T(a;1)

Question: can "a" = "a'" , thus forming a loop?

Now we know a = 4i+3 --> a' = 6i+5

cond: a' - a = 0 = (6i+5) - (4i+3) = 2i + 2 = i + 1 --> i = -1

So we know, that the number

a = 4 i + 3 = -1

transforms to

a' = 6 i + 5 = -1

in one primitive step.

Since here I discuss only positive integers as members of the collatz-transformation, this value is not allowed, thus the one-step-loop with exponent 1 is disproven on a modular basis.

This functions analoguously with members of other residue classes.

Let's try class C2 with A = 2

a' = T(a; 2)

This requires a to be of the form

a = 8 i + 1 a' = 6 i + 1

a' - a = 0 = 2i

Only solution is

i = 0 ---> (a->a' ) = (1 --> 1)

and no other solution

Let's try class C3 with A = 3

a' = T(a; 3)

This requires a to be of the form

a = 16 i + 13 a' = 6 i + 5

a' - a = 0 = 10i + 8

No positive nor integral negative solution possible.

For all other classes the result is the same as for class C3.

Now we look at a 2-step transformation

a ' = T(a;1,A)

To make things better visible, we part that into two single steps

b = T(a;1) c = T(b;A)

setting A = 1

a must be of the form 4i+3, then b is of the form 6i+5 .

Then b must also be of the form b = 4j+3 and c is of the form 6j+5.

To form a loop, a must satisfy two conditions:

a = 4i+3 and a = 6j+5

This says,

from a = c: 4i + 3 = 6j + 5

from b = b: 6i + 5 = 4j + 3

equalling cofactors of i

from a = c: 12i + 9 =18j + 15

from b = b: 12i +10 = 8j + 6

subtracting

-1 =10j + 9

j = -1

i = (6j+2)/ 4 = -4/4 = -1

following a = -1 b = -1

and

-1 = T(-1;1,1)

a solution, which is not valid for the original Collatz-problem.

setting A = 2

a must be of the form 4i+3, then b is of the form 6i+5 .

Then b must also be of the form b = 8j+1 and a is of the form 6j+1.

To form a loop, a must satisfy two conditions:

a = 4i+3 and a = 6j+1

b = 6i+5 and b = 8j+1

This says,

from a = c: 4i + 3 = 6j + 1

from b = b: 6i + 5 = 8j + 1

equalling cofactors of i

from a = c: 12i + 9 =18j + 3

from b = b: 12i +10 =16j + 2

subtracting

-1 = 2j + 1

j = -1

i = (6j-2)/ 4 = -8/4 = -2

following b = -7 a = -5

and

-5 = T(-5;1,1)

a solution, which again is not valid for the original Collatz-problem.

==> doc_lg32_approx(550,3,true)

.

1 1+ 3.3333333e-0001 1.0000000e+0000

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