Collatz-Intro - Approximations of log(3)/log(2): Some illustrative graphs


Gottfried Helms Univ. Kassel

mailto: helms at uni-kassel

www: math-homepage



In the following I present some graphs, which illustrate a known critical condition, which serves for primitive loops (or "1-cycles") as well as for the "waring-problem".

The condition is given as


Let b=3 and x = b^N = 3^N be displayed as a irregular fraction base a power of 2:


let q = 2^N   


  and d be the integer




then let x be represented as


x = d * q  + r   


which means, that in the numbersystem base(q) the both digits d and r make x


  x = "d r"  (digits d,r base(q))



The critical condition for a 1-cycle (as well for the waring-inequality) is then



  d + r < q 



because it is needed, that by division by q-1 no carry occurs in


 x      "d r"   "d r"   "d r"
 --- =  ----- + ----- + ----- + ... = "d r" + "d.r" + "0.d r" + "0. 0 d r" + ...
 q-1      q      q^2     q^3


   = " d d.d d d d ..." + " r.r r r r r..."


   = " d d.d d d d ..."
     + " r.r r r r r..."



  x - 1      " d d.d d d d ..."
  ----- = +  " 0 r.r r r r r..."
  q - 1   -  " 0 1.1 1 1 1 1..."




Now the critical condition for a primitive loop says, that the result must


 a) be integer

 b) a power of 2


to make a primitive loop possible since


       3^N - 1      x - 1
2^A = ---------- = -------   ( the rhs is just the reformulation above)
       2^N - 1      q - 1


In the base q-system this means:

  e 0 0 =   " d d.d d d d ..."
          + " 0 r.r r r r r..."
          - " 0 1.1 1 1 1 1..."      (  where  e =2^A)


Since d is always smaller than e, one can immediately see, that it must be e-1; also each digit of the rhs-sum must produce a carry , and the new resulting digit must precisely 0.

So this means, to have a "primitive loop", it needs, that d+r = q  or a multiple d+r = i*q.

Therefore it is a most interesting question, whether


  d + r < q 


holds for the given rhs-term for all N.


If this statement could be proven for the interesting values of x and q, then the primitive loop is disproven as well as the "waring-problem is completely solved" (see [mathworld/waringproblem] and    mathworld/powerfrac ) .


The same problem was attacked by Kurt Mahler, who analyzed that question systematically and formulated a criterion called Z-numbers. If there were no Z-numbers, then this would be equivalent  to the proof of the critical condition. Mahler didn't succeed in proving the nonexistence of such Z-numbers, but could at least state, that at most finitely many such numbers exist. (see [mathworld/Z-Numbers])




Below the graphs, which may illustrate that problem.





Data x = 3^N are shown in a two-dimensional table, where the y-coordinate is d and the x-coordinate is r.

If the sum of  d+r < q for all interesting x then this means, that all entries are below the main diagonal.


Normally one would expect, that the value of r is randomly and uniformly distributed and independend of d. But there are also exceptions, as it seems with the combination of x = 3^N.


In the following I show also some variations of the problem, where instead of basis=3 some other bases=b in the range 2 < b < 4 were taken to illustrate specific variations of the problem.


In the following graphs the areas of higher N were mapped into the same table by rescaling. So the points of the scatter must indicate the power N; I used a bigger size for lower N and smaller size for higher N.




Fig. 1: an arbitrary base b other than 3 exhibits an expected random-scatter on the x-axis (=r), independent from the y-value (=d). In this case I took log(64)/log(3), since it is a special pretty case for this claim :





One can see, that no obvious relation is between the y and x-values; that means for each d there is an arbitrary r in


 x/q = d + r/q   with x = b^N and q = 2^N






Fig. 2: the picture for b=3 exhibits, that the sum of d+ r is always smaller than q, or  r< q-d:





This picture means, that empirically the condition holds above N=1 up to N=64 ( ;-) a *small* value indeed) and it easily could be, that even that restriction could be formulated sharper, as we see, that always a certain distance remains to the diagonal, where the best approximations may hyperbolic or logarithmic decrease with a function of N.




Since the condition d + r < q steming from the Collatz-loop-discussion was identically formulated in the analysis of the Waring-problem, but is unproven yet, it is of interest to see how variations of the problem behave. From an analysis (not finished yet) of the approximation-ratio as a function of N there is a strong indication, that this problem may be related to the properties of the number phi (golden ratio : phi - 1 = 1/phi   or phi = (1+sqrt(5))/2  ), and phi could play a role as a certain critical value in that approximation, as a "norm" it may behave like a zero-element.

Indeed we see a beautiful coincidence between m and p in the table for phi:



Fig. 3: the picture for b=2*phi exhibits, that phi certainly plays a special role in the approximation-analysis:





This is a very remarkable illustration, since here we do not need approximation-estimations. Every 2'nd value N exhibits either the smallest or the highest residue of a class, something like d + r = c/N like an exact value, depending on N.

Clearly this comes from the property of phi, that


 phi^2 = phi^1 + 1


which means, that the fractional part of phi^2 and phi is the same.


This basis phi is the only case I can imagine, where the possibility of determining a function for the quality of an approximation depending on N seems given- in all other cases that function would be stepwise or somehow else erratic, with good and bad approximations for increasing N - and only a general bound for the intermediate best approximations may be accessible as a function on N, like the one, that is formulated by the waring-condition



Now take phi as a kind of norm-value. What does happen, if we slightly change this value?


Fig. 4: here phi was modified by simply cutting digits to 1e-6:





One can see, that for higher values N this modification comes into play, but still a certain regularity remains at the beginning



Fig. 5: another slightly distortion of the value of phi




The generally expected "randomness" for r (=x-coordinate) occurs from higher N.



In the following some randomly chosen values around the interesting base 3:


Fig. 6: some deltas added to 3





The  d + r < q  condition is already spoiled, at least for small values of N.




















Another interesting question is, whether some other known constants do show a typical regularity, from where approximation-rules could be derived. The 2-log of e has an impressive pattern:


Fig. 7:





This number seems to have the same property as the basis 3, and even it seems to have a better regularity in the pattern as well as a greater distance to the d+r < q - diagonal.



Variations over PI


Fig. 8.1-1: Variations on pi^2





This number seems to have a hyperbolic characteristic. This value seems to satisfy that  d + r < q very good, much more than the 3^N/2^N case, which is relevant for the loop-problem of the Collatz-transformation.


Fig. 8.1-2:





This one not.


Fig. 8.1-3:





But this one again.



Fig. 8.2: Some variations about pi^3





This number seems to have a hyperbolic characteristic as well, but near approximations seem to occur with higher values of d (see red dot in the right below corner).


Fig. 8.2-2: Some variations about pi^3





Only some combinations of base b=(pi^A/2^B) show this hyperbolic behave. Here  is d +r < q , and much more extreme than in the original base b=3 (=3^N/2^N)-matter




Fig. 8.3-1: Even some powers show exotic behave





Only some combinations of  b=(p^A/2^B) show this hyperbolic behave. Again, that may be a candidate for further investigation of its behave regarding d + r < q, which seems even more clear than in that original one with b=3 ( 3^N/2^N- problem) .




It may be, that if one takes bases differing from b=3 (as in the collatz-problem) like the ones, which were shown here, one could make similar observations (or completely different) to that of the 3x+1-problem, especially in regard to the question of loops.


Currently I am investigating the approximation-behavior of the 3^N/2^N-problem, and may be from that an argument for further restrictions for a N-dependent-formula for that approximation-quality can be found, which empirically seems much higher than the waring/primitive-loop-criterion, thus denying the primitive loop on a much higher level.


(My current best guess for the highest low-bound is something of this function of N

(current guess)


          1             powerceil2(3^N)
   1+ ------------  <= ----------------  ,
       10 N log(N)          3^N



 which I estimated from numerical and graphical display for values up to some N~10^40.)



Gottfried Helms