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 ns are
each successive calculated extension 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.
My Collatz Home Page
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