Collatz-Intro - Some Log3/Log2 observations

Workshop
Recreational
Mathematics

Gottfried Helms Univ. Kassel

The Log(3)/Log(2) problem occurs everywhere in the collatz-problem, but most directly in the discussion of the primitive loop.

One current example is the connection to the Waring-Problem [mathworld/waringproblem], where I found the inequality below.

The reader my wonder, why this chapter is titled "log(3)/log(2)" and does not find any logarithm here...

Number-theory problems like these are mostly expressed in log's, for several reasons:

· it may sometimes provide a more general framework,

· it may be more concise, and surely,

· it may be, because the actual numerical computations are sometimes much easier if one uses the "logarithm-times-N" instead of "base-to-the-power of N".

But for clarity and readability I favor in this case to express things in the form of powers.

"The Waring-problem can completely solved,..." if the the inequality [W1] below can be proven. (according to Weisstein, mathworld.com).

[W1]: critical inequality from the Waring-problem:

                3^N          3^N
          frac( --- ) < 1 - ---
                2^N          4^N

In the article about "Power Fractions" Weisstein then states the next inequality [W2]as an extension of this problem. [extended inequality] , where the left inequality is also mentioned as still unproven:

[W2]: extended inequality from the Waring problem

   3^N          3^N          3^N
   --- < frac( --- ) < 1 - ---
   4^N          2^N          4^N

To relate this inequality to that of my "critical inequality for loops", I have to bring them in a corresponding form resp. their denominators and the fraction()/powerceil()-terms.

Since I have done the loop-approximation always with the ceil(), resp. powerceil2() function, I convert the frac()-expression into a ceil()-expression by the following equality:

frac(x) = x - floor(x) = x - ceil(x) +1

and then I get for the extended Waring-inequality this intermediate form:

[W3]:

3^N 3^N 3^N 2^N 3^N
--- + --- < ceil( --- ) < 1 + --- - ---
2^N 4^N 2^N 3^N 4^N

Again, since I did empirical approximation using 3^n in the denominator, getting the theoretical range of 1..2 for the recomputed ceil()-term, I adapt the above formula to my convention:

multiplying by (2/3)^N, changes also the ceil()-term to powerceil2()-term

[W4]:

1 powerceil2(3^N) 2^N 1
1 + --- < --------------- < 1 + --- - ---
2^N 3^N 3^N 2^N

This is obviously an enhancement of the widest and intuitive bounds

               powerceil2(3^N)
1         < --------------- < 1 + 1
                     3^N

Now I relate [W4] to my condition of the critical inequality of the primitive loop.

This is, (putting the free parameter i to its worst case value i=1 in that formula, makes it vanishing here)

[H1]:

3^N-1
powerwceil(3^N) <= 2^N* -------
2^N-1

[H2]:

powerwceil(3^N) 2^N 3^N-1
--------------- <= ---* -------
3^N 3^N 2^N-1

In this the lhs must be smaller or equal to make a primitive loop possible.

Now I compare the rhs of this inequality to the lhs of the Waring inequality [W4]. It's a very similar term.

I expand the 3^N-1- fraction in a series:

3^N-1 3^N-1 3^N-1 3^N-1 3^N-1 3^N-1
------- = ------- + ------- + ------- + ------- + ------- +
2^N-1 2^N 4^N 8^N 16^N 32^N

getting

2^N   3^N-1            1      1     1      1
---* ------- =   1 + ---- + --- + ---- + ---- + ... +
3^N   2^N-1          2^N    4^N    8^N   16^N

                       1     1     1      1
                    - ---- - --- - ---- - ---- - ... - (red marked term added)
                      3^N    6^N   12^N   24^N           (correction 18.8.04)

Adding the negative Elements pairwise to that of the positive elements with the nearest smaller denominator, then it come out to be:

2^N 3^N-1 1 3^N - (2^N+1) 1

---* ------- = 1 + ---- - -------------- = 1 + --- - eps (correction, was "+ eps")

3^N 2^N-1 2^N 3^N * (4^N-1) 2^N

where eps is smaller than the square of the last term, and the whole rhs-term is slightly smaller than the lhs-term in the extended Waring-inequality [W4]

This makes my critical inequality [H2] look like

[H3]: a primitive loop can only exist if the inequality would hold.

powerceil2(3^N)          1
--------------- <= 1 + ---     - eps
     3^N                2^N

which would allow a primitive loop (from the range-condition only).

In the reverse focus of a *denying* of the primitive loop, this inequality could be turned around:

[H4]: a primitive loop of length N is denied, if the inequality holds .

        1             powerceil2(3^N)
   1 + --- - eps <= ---------------
       2^N                3^N

and this is practically the lhs Waring-inequality [W4] with a slightly smaller, thus weaker low-bound. The lhs of [W4] corresponds with the rhs in [W2] and thus to the original Waring-inequality.

Putting [H4] in in whole numbers by multiplying by 3^N then eps will be smaller than 1, thus a neglectible quantity in the comparision of the both formulas.

Now the nonexistence of the primitive loop *is* already proven [Ray Steiner], [Simons/De Weger]by approximation arguments similar to that arguments here.

So -if arguments of R. Steiner's proof are equivalent to my critical inequality - then I think,

· that this proof in turn fixes the low-bound in the Waring-inequality , too, by the near-identity of the critical equations.