CollatzIntro  Approximations of log(3)/log(2): Some illustrative graphs 
Workshop 

In the following I present some graphs, which illustrate a known critical condition, which serves for primitive loops (or "1cycles") as well as for the "waringproblem". 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
d=floor(x/q) 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 1cycle (as well for the waringinequality) is then d + r < q because it is needed, that by division by q1 no carry occurs in x "d r" "d r" "d r" = " d d.d d d d ..." + " r.r r r r r..." = " d d.d d d
d ..." and x  1 " d d.d d d d ..."
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 In the base qsystem this means: e 0 0 = " d d.d d d d ..." Since d is always smaller than e, one can immediately see, that it must be e1; also each digit of the rhssum 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 rhsterm 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 "waringproblem 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 Znumbers. If there were no Znumbers, then this would be equivalent to the proof of the critical condition. Mahler didn't succeed in proving the nonexistence of such Znumbers, but could at least state, that at most finitely many such numbers exist. (see [mathworld/ZNumbers]) 
Below the graphs, which may illustrate that problem. 
Legend: Data x = 3^N are shown in a twodimensional table, where the ycoordinate is d and the xcoordinate 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 randomscatter on the xaxis (=r), independent from the yvalue (=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 xvalues; 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< qd: 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 Collatzloopdiscussion was identically formulated in the analysis of the Waringproblem, but is unproven yet, it is of interest to see how variations of the problem behave. From an analysis (not finished yet) of the approximationratio 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 zeroelement. 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 approximationanalysis: This is a very remarkable illustration, since here we do not need approximationestimations. 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 waringcondition 
Now take phi as a kind of normvalue. What does happen, if we slightly change this value? 
Fig. 4: here phi was modified by simply cutting digits to 1e6: 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 (=xcoordinate) 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 approximationrules could be derived. The 2log 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. 8.11: 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 loopproblem of the Collatztransformation. 
Fig. 8.12: This one not. 
Fig. 8.13: 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. 8.22: 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. 8.31: 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 collatzproblem) like the ones, which were shown here, one could make similar observations (or completely different) to that of the 3x+1problem, especially in regard to the question of loops. Currently I am investigating the approximationbehavior of the 3^N/2^Nproblem, and may be from that an argument for further restrictions for a Ndependentformula for that approximationquality can be found, which empirically seems much higher than the waring/primitiveloopcriterion, thus denying the primitive loop on a much higher level. (My current best guess for the highest lowbound is something of this function of N
which I estimated from numerical and graphical display for values up to some N~10^40.) 
Gottfried Helms
27.8.2004