What is Dull (Interesting?) About the Extensions?

I have said that the extensions are dull and uninteresting compared to the action which occurs in the left descents.  But it may be worthwhile to be more specific and probe more deeply. 

The series of extensions from any given left descent element may be calculated iteratively by applying 4n+1 where the first n is the left descent element parental to the extensions and successive extensions are itertively calculated in turn.  To show the dull behavior,  we can look at the base 4 logarithm of any series of extension elements.  One expects that the mantissas will change little due to the addition of 1 at each iteration,  because little effect ensues when the extensions become large.  I'll illustrate with the series from 1, 103, and 1003.  The former will show the largest drift in the mantissa and the latter will show a much smaller drift.  By the time the extension is between 4^12 and 4^13,  the mantissa is changing only in the eighth decimal place.  The family of curves defined by these base 4 logs will clearly never cross -- the mantissas are increased by the addition of 1 at each additional extension but by a smaller and smaller increment as the magnitude of the extensions increases. 

    parent &      log[4]     parent &     log[4]     parent &    log[4]
   extensions  (extension)  extensions  (extension) extensions (extension)

          1     0.00000000
          5     1.16096405
         21     2.19615871
         85     3.20469547       103    3.34325026
        341     4.20681397       413    4.34499899      1003    4.98505294
       1365     5.20734262      1653    5.34543551      4013    5.98523272
       5461     6.20747472      6613    6.34554459     16053    6.98527766
      21845     7.20750774     26453    7.34557186     64213    7.98528889
      87381     8.20751600    105813    8.34557868    256853    8.98529170
     349525     9.20751806    423253    9.34558038   1027413    9.98529240
    1398101    10.20751858   1693013   10.34558081   4109653   10.98529258
    5592405    11.20751871   6772053   11.34558092  16438613   11.98528262
   22369621    12.20751874  27088213   12.34558094  65754453   12.98529263

It remains to indicate how the series of extensions interleave as larger and larger parental left descent elements are encountered.  A program was written in MapleV4 to determine the base 4 mantissas at 4^12 for all parents from 1 to 4^5.  The output from that program was sorted in increasing order of the mantissas and a segment from 47 to 49 appears below.  In the pass through the first order of magnitude (base 4,  of course) above 47 to 49,  the numbers 191,  193,  and 195 are inserted,  and in the next order of magnitude above them,  three additional insertions occur in turn between each of those numbers.  The mantissas (at 4^12,  after the values have settled to 7 decimal places) are approximately equally spaced,  though the spacing decreases slightly as the magnitude of the extensions increase.  The last column gives the difference between the first mantissa,  that of the parent,  and the mantissa at 4^12; this reinforces the observation that the added 1 causes a larger drift in the mantissa for smaller parents. 

The table shows a region where no parents are less than 4^3,  so shown parents are in the range of 4^3 to 4^5.  Understand that 47,  189,  and 757 are in the same series of extensions,  all represented by the left descent element parental to the whole series.  The spacing of 16 between the 4^3 parents and of 4 between the 4^4 parents is clearly a consequence of the way each successive pass of the parents through the base 4 orders of magnitude must interpolate its series of extensions among those visited in earlier passes. 

            log[4]        mantissa of     extension     delta of
parent     of parent     log[4] at 4^12    at 4^12     mantissas
   47    2.777294425838  .782392304547    49632597  .005097878708
  759    4.783978037707  .784294760661    49763669  .000316722953
  761    4.785876321751  .786192212501    49894741  .000315890749
  763    4.787769623417  .788084686325    50025813  .000315062907
  191    3.788714414017  .789972208183    50156885  .001257794165
  767    4.791541383751  .791854803924    50287957  .000313420172
  769    4.793419893980  .793732499193    50419029  .000312605212
  771    4.795293524957  .795605319436    50550101  .000311794478
  193    3.796228518634  .797473289902    50681173  .001244771268
  775    4.799026250080  .799336435643    50812245  .000310185563
  777    4.800885394203  .801194781519    50943317  .000309387315
  779    4.802739759030  .803048352197    51074389  .000308593166
  195    3.803665156874  .804897172154    51205461  .001232015279
  783    4.806434248645  .806741265681    51336533  .000307017036
  785    4.808274421889  .808580656882    51467605  .000306234993
  787    4.810109912753  .810415369678    51598677  .000305456924
   49    2.807354922057  .812245427805    51729749  .004890505748

One could argue that this systematic distribution of the series of extensions across the region between successive powers of 4 is largely responsible for the Collatz predecessor trees managing to reach all the integers.  So while the extensions may be regarded as dull,  their dull and systematic behavior is undoubtedly key to the overall behavior which the conjecture postulates. 

Since every (odd integer) parent to a series of extensions maps to a particular precise mantissa (take any precision you like),  each extension series may be characterized uniquely by the odd integer which is its parent.  Thus,  the distribution of base 4 mantissas gives insight into the manner in which the odd integers of the extensions will be arrayed once the abstract generation tree has identified each parental odd integer as the member of some left descent.

A Parallel View using the Predecessor Tree Including the Even Integers

The above discussion is presented from the world view that the Collatz predecessor tree is most revealingly examined if only the odd integers encountered in the iterations are included.  It is this view which leads to the drift in the mantissas of the logarithms as 1 is added each time an extension is added to its series.  But what if,  for a moment,  we include even numbers in the Collatz predecessor tree?   I am indebted to Joseph Parranto for an e-mail discussion which led to this altered viewpoint and where it leads.

In a predecessor tree which includes the even numbers we might arbitrarily assign the whole right descent composed of the values
           n*2i (i = 0..infinity)
to each odd n,  to avoid having those right descent elements repeat for 2n,  4n,  8n,  etc.  Looked at in this way,  the emerging picture is very like that described above.   The right descents arise from every odd integer  (like the extensions above)  but the contents are developed at each stage by multiplying by 2 instead of by multiplying by 4 and adding 1.   In both cases the right descents go off to infinity.

In this view,  there is no drift in the value of the mantissas as larger and larger numbers in the series are reached,  and use of log2 seems a better choice than log4 for demonstrating the controlled interleaving of the odd integers such that every odd integer  (and their multiples of powers of 2)  would eventually be found.   No table could represent the situation as powerfully as a picture prepared using the graphing functions of MapleV4.

In this polar plot the odd integers from 3 to 255 are represented by radii.  Each radius appears at an angle defined by 2*Pi*mantissa of the log2 of its integer,  and the inner end of the radius appears at the characteristic of the log2 of its integer,  with the outer end of every radius at 8.  The crowded nature of the plot makes it impossible to label more than a very few of the radii with their represented values.

Clearly,  the odd integers  (and,  of course,  all their multiples by powers of 2)   will continue to interleave in the illustrated way so that all of them will have a place in such a picture.  Were we to show that all the odd positive integers have a place in the Collatz predecessor tree,  it immediately follows that all the even positive integers do also.

An illustration of a particular Collatz trajectory path in the same kind of polar plot gives a look at the changing characteristics and mantissas of the base 2 log of the integers in the trajectory.

There's a paradox lurking here, but I'll leave it for later.


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