Re: The difference table of pures in the 3n+1 problem!

On Apr 29, 11:17 pm, Danny <fasttrac...@xxxxxxxxxxxxx> wrote:
On 29 Apr, 00:09, Danny <fasttrac...@xxxxxxxxxxxxx> wrote:





Looking at the mod 18 table and then
making a comparison with the differences
between the pures I cam up with this
table.

Where no row can sum more or
less than 18 except row 1.

The table order for each row after row 1
is [1,2,3,1,2,3,3,3]

The pures difference table showing the disruptions
in the pattern created by the non trivial impures .

3,1,2,3,3,3
1,2,3,1,2,3,3,3
1,2,3,1,2,3,3,3
1,2,3,3,3,3,3---where 61 changes order for this row.
1,2,3,1,2,3,3,3
3,3,1,2,3,3,3---where 91 changes the order.
1,2,3,1,2,3,3,3
1,2,3,1,2,3,3,3
1,2,3,1,2,3,3,3
1,2,3,1,2,3,3,3
1,2,3,1,2,3,3,3
1,2,3,3,3,3,3---where 205 changes the order.
1,2,3,1,2,3,3,3
1,2,3,1,2,3,3,3
3,3,1,2,3,3,3---where 253 changes the order.
1,2,3,1,2,3,3,3
1,2,3,1,2,3,3,3
1,2,3,1,2,3,3,3
3,3,1,2,3,3,3---where 325 changes the order.
1,2,3,1,2,3,3,3
1,2,3,1,2,3,3,3
1,2,3,1,2,3,3,3
1,2,3,1,2,3,3,3
3,3,1,2,3,3,3---where 415 changes the order.
3,3,1,2,3,3,3---where 433 changes the order.
1,2,3,1,2,3,3,3
1,2,3,1,2,3,3,3
1,2,3,1,2,3,3,3
1,2,3,1,2,3,3,3
1,2,3,1,2,3,3,3
1,2,3,1,2,3,3,3
1,2,3,1,2,3,3,3
3,3,1,2,3,3,3---where 577 changes the order.
1,2,3,1,2,3,3,3
1,2,3,1,2,3,3,3
1,2,3,3,3,3,3---where 637 changes the order.
1,2,3,1,2,3,3,3
1,2,3,1,2,3,3,3
1,2,3,1,2,3,3,3
1,2,3,1,2,3,3,3
1,2,3,1,2,3,3,3
3,3,1,2,3,3,3---where 739 changes the order.
etc..
Note the different pattern between a 1(mod 18)
exception and a 7(mod 18) exception.

A nice comparison with the mod 18
table that shows only the impure exceptions
that alters the pattern of only the row the
exceptions reside on. Which is the same
as above with the difference table of pures
where only the odd exceptions (impures)
change the order for that row it resides on!

Also supporting the fact that only the non
trivial odd exceptions are the most important
hailstones in the Collatz 3n+1 problem
making all other hailstones trivial because
they are part of a fixed pattern that are
in 16 columns that never change in the
mod 18 table.
Whereas the other 2 columns, odd 1(mod 18)
and odd 7(mod 18) will have changes at times
with these non trivial odd exceptions(impures).

A little more experimenting with these non trivial
odd exceptions I found that the smallest seed
possible that creates the non trivial hailstone
has to be pure itself to create the non trivial
hailstone.

Also experimenting with seed paths and recording
the mod 18 status of each path member is also
interesting were a series of ..17,16,17,16,17,16,8 --
mod 18 is just an even \ odd cycle of (3n+1)/2
and (8) is just the second sequental (/2) incountered
in this series of 17,16,17,16..

Dan

Just an example of a typical seed and its sequence members
applying the mod 18 to each member.

The (**) are the integer members of the
two non trivial columns in the mod 18 table
of 1(mod 18) and 7(mod 18)

Seed 259109

777328 == 16 (Mod 18)
388664 == 8 (Mod 18)
194332 == 4 (Mod 18)
97166 == 2 (Mod 18)
48583 == 1 (Mod 18)**
145750 == 4 (Mod 18)
72875 == 11 (Mod 18)
218626 == 16 (Mod 18)
109313 == 17 (Mod 18)
327940 == 16 (Mod 18)
163970 == 8 (Mod 18)
81985 == 13 (Mod 18)
245956 == 4 (Mod 18)
122978 == 2 (Mod 18)
61489 == 1 (Mod 18)**
184468 == 4 (Mod 18)
92234 == 2 (Mod 18)
46117 == 1 (Mod 18)**
138352 == 4 (Mod 18)
69176 == 2 (Mod 18)
34588 == 10 (Mod 18)
17294 == 14 (Mod 18)
8647 == 7 (Mod 18)**

Where the 7(mod 18) always has to have the
preceeding residuals of 4,2,10,14 in that order
whereas the 1(mod 18), only requires at a
minimum, preceeding residuals of 4,2.

Probably a good reason why there are so many
more 1(mod 18) impures then 7(mod 18) impures.

Dan

Just a quick note, I'm not ignoring you. I just haven't had
any time to look at this latest info. I've been studying Ken
Conrow's Left Descent Assemblies, translating his Maple
code to Python and working out the relationship between
Ken's LDAs and my Sequence Vector system.

Turns out the LDAs are a Sequence Vector subset and
my tools for analyzing Sequence Vectors can also be
employed to solve certain LDA questions.

Supposedly, my Google Group TrueButUnproven will let
me store files for download in addition to messages. I'm
going to take my Collatz library, add copiuos commentary
so it can be translated to other languages and make it
available from my group along with some write-ups and
examples.

While that's going on, I'll see if I can find time to look over
this thread.

.

Relevant Pages

  • Re: The difference table of pures in the 3n+1 problem!
    ... The pures difference table showing the disruptions ... table that shows only the impure exceptions ... trivial odd exceptions are the most important ... applying the mod 18 to each member. ...
    (sci.math)
  • The difference table of pures in the 3n+1 problem!
    ... The pures difference table showing the disruptions ... table that shows only the impure exceptions ... trivial odd exceptions are the most important ... making all other hailstones trivial because ...
    (sci.math)
  • The difference table of pures in the 3n+1 problem!
    ... The pures difference table showing the disruptions ... table that shows only the impure exceptions ... trivial odd exceptions are the most important ... making all other hailstones trivial because ...
    (sci.math)
  • Re: What is next in the series? 2 more Q
    ... I can invoke Occam's razor to get yet another simple answer. ... The pattern is odd, even, odd, even, ... ... decrease by 2 and the even numbers decrease by 36. ...
    (sci.math)
  • Re: english is unstructured?
    ... 'Arthurian' is 'ArTHURian'. ... The OP's complaint was that the stress was random. ... there is a pattern common to many English words. ... With all these exceptions: ...
    (alt.usage.english)