Welcome
mensa...@aol.com
Conjecture. Hopefully, I'll have some stuff here you won't
find elsewhere on the net. We'll see how it goes.
Ernst
I think this is over due.
I'm glad we have a group to chat about Collatz..
I'm Ernst and having a forum where the messages don't scroll away into
the distance will be nice.
Ernst
You want to work with me to develop an attractor program?
I mean I have one but I bet it could stand a rewrite.
mensa...@aol.com
Could you elaborate a liitle? I've got one also for 3n+C.
Ernst
I'm considering a Open Source rewrite.
I could stand to work with other programmers.
So you have your own? Did you publish it?
mensa...@aol.com
Not yet. That's what I was planning to do here.
The [FAQ] articles are intended to provide some
background so that the Python programs will make
some sense. And when I say "program", what I have
is a library of utility functions.
For example, I have a utility called collatz_C that
generates a Collatz sequence for any 3n+C system
that runs until it detects that the sequence enters
a loop at which point it returns the number at which
it entered the loop, loop size in terms of iteration
counts and the attractor (smallest odd number in the
loop cycle).
For an example of how I use that utility, let's say I
am looking for an example of a very long loop.
from collatz_functions import *
lmax=0
c = 3
while c<10000000:
c = gmpy.next_prime(c)
# primes make good candidates for long loops
a = collatz_C(1,c,0)
# every 3n+C sequence beginning with 1 ends in
# a non-trivial loop (although 1 itself may or
# may not be a member of the loop)
if c==a[3]: # a[3] is the attractor
# print 'c:%4d l:%4d ' % (c,a[3]),
# print ' <--- 1 falls into trivial loop'
pass
else:
if a[5]>lmax: # a[5]is the loop size
print 'c:%12d l:%12d ' % (c,a[3]),
sv = build_sv(a[3],a[3],999999,c)
# print sv
lmax = a[5]
print lmax,sum(sv)
This program will search for longer and longer loops based
on 1 as the seed. From it, I got
3n+1271069 has an attractor at 97 whose sequence contains 315529
numbers (115819 odd and 199710 even).
Is that kind of what you had in mind? I can put off the
[FAQ]s for a while and start posting my program code.
Or you can post what you're doing in your group and
I'll have a look at it. We use different languages, but
we should be able to figure out each others algorithms.
I don't know if it is possible to predict where loops will
be found (any more than you can predict what a number's
factors are), so you always have to search and test.
Sometimes it's not so much the program as the way you
apply it. Using seed=1 C=prime is an example. I still
have to search, but it helps to know where to look.
Or where not to look. Any brute force search algorithm
can be improved if you exclude places where solutions
cannot exist. I've got other techniques for that also.
All of which I plan to document here.
Ernst
I'm trying to motivate. I have spent some time away from the math.
The crux on my program is like yours.. Looks for a repeat in the
stream and keeps track of the lowest value.
You know I used to get so up tight about this.. I don't feel that way
anymore.
I feel good.
Maybe the thing to do first is trap a few more stats?
I first shared the attractor info with a guy in Canada. He varified
my data whithout needing my program.
He said he used some buffering where if a path has been seen the
sequence aborts thus speeding up the processing of many values.
I guess I could do that to mine.. Do you have that in yours?
mensa...@aol.com
Ernst wrote:
> Oh no Do what you like!
I am. My latest [FAQ] is Factor Congruence, the principles of
which will play an important part in what I post next. I skipped
the [FAQ] on the derivation of the Crossover Point Function for
now to get to the stuff about attractors. But it is stated in
[FAQ] Factor Congruence and that's sufficient for now as the
parameters and their derivation are given in [FAQ] Hailstone
Function.
>
> I'm trying to motivate. I have spent some time away from the math.
>
> The crux on my program is like yours.. Looks for a repeat in the
> stream and keeps track of the lowest value.
Although you must ignore the lowest value until you are sure you're
inside the loop (as I found out the hard way).
>
> You know I used to get so up tight about this.. I don't feel that
way
> anymore.
>
> I feel good.
But you still need to open up a little more. To date, you haven't
actually said _anything_ about your attractor program. And I don't
mean post the source code. What I'm trying in my [FAQ] articles is
to discuss principles which will then later be implemented in code.
Don't assume anyone knows what you're talking about. For instance,
you don't have to explain the Collatz Conjecture, there are plenty
of web sites where readers can look it up. You'll notice my [FAQ]
on the conjecture itself is just a couple links.
But when you bring up the idea of "attractor", you have to say
what it is and the context in which it is used. Use links if
need be, even to my articles if they would help.
Then, explain in general terms what your "attractor program" is
trying to accomplish. For instance, mine is simply a utility
to identify the attractor (there will always be one) in a 3n+C
system. Now you can wrap that utility inside a larger program
that specifically searches for attractors as in my example
program that searches only seed=1 C=prime.
I _do_ have a program that searches all possible Sequence Vectors
for integer Crossover Points (see [FAQ] Factor Congruence). That
identifies attractors without needing to ever run a Collatz
sequence. The upcoming article
Factor Congruence - a case study
will compare the two methods of identifying attractors.
>
> Maybe the thing to do first is trap a few more stats?
In my planned article
Factor Congruence - a case study
there will be a _lot_ of stats. It might take me a while to
put together and will probably be quite long as there is so much
data. Probably more than is necessary, but keep in mind I'm
learning this stuff as I go along and I like extreme confirmation
of my ideas.
>
> I first shared the attractor info with a guy in Canada. He varified
> my data whithout needing my program.
Well, that's ok, we don't need to see it right now. But at some
point if your program is not doing all that you want it to and
want some help rewriting it, you will have to post _something_.
My main motivation for this site is that by forcing myself to
explain things, I often see places where more research is needed.
So it doesn't really matter if no one is reading my articles, it
helps me and that's what's important.
> He said he used some buffering where if a path has been seen the
> sequence aborts thus speeding up the processing of many values.
> I guess I could do that to mine.. Do you have that in yours?
No. I assume by that he means when he starts a sequence at 31,
he recognizes that he's on the same path as when he started from
27 and doesn't need to run all the way to the end. My utility
doesn't track anything from one sequence to another but I suppose
that could be added (depending on memory/time overhead). I did
something like that when searching for large loops, tracked how
big the loop sequence was. If it was larger, then it must be a
loop I hadn't seen before.
Trouble is, stats alone only take you so far. I'm trying to learn
the underlying principles that control what's going on. If I can
get a handle on that, then I don't need any more stats. I can
focus my programs on exploiting those principles as is the case
with the Sequence Vector searching.
Ernst
<sigh> I'm trying to get there...
I dug out the backups and started putting files back online today. I
didn't find the last CD.. If I don't find that last CD I'll have to
rewrite that latest development I have spoke about. Bummer. You must
know how many hours of testing can go into a program.
Then again It could work out better for sharing later if I start out
knowing it works and write clearly. Unlike that last time where i
didn't know if it worked and just hacked old versions till I knew.
Yeah I don't have anything to post yet so I have to stand by and read
what you put up...
I think I will keep it simple for now.. I don't feel that driving
energy like I did before.
mensa...@aol.com
Ernst wrote:
> It looks like you are motivated...
>
> <sigh> I'm trying to get there...
>
> I dug out the backups and started putting files back online today.
I
> didn't find the last CD.. If I don't find that last CD I'll have to
> rewrite that latest development I have spoke about. Bummer. You
must
> know how many hours of testing can go into a program.
> Then again It could work out better for sharing later if I start out
> knowing it works and write clearly. Unlike that last time where i
> didn't know if it worked and just hacked old versions till I knew.
>
>
> Yeah I don't have anything to post yet so I have to stand by and
read
> what you put up...
Ok, check out my latest post, Factor Congruence - a case study.
Remember, it's all about having fun.
Ernst
Are you talking about c ? where c is > 0 and c is odd ?
Or are you talking about c = 1 ?
I admit I have a lot to catch up on.
mensa...@aol.com
Ernst wrote:
> Factor congruence
>
> Are you talking about c ? where c is > 0 and c is odd ?
> Or are you talking about c = 1 ?
any congruence matching, which means there won't
be any non-trivial attractors in 3n+1 since all
non-trivial attractors depend on that factor
congruence to form loops.
That almost sounds like a proof of 3n+1 but not quite.
You see, if C has factors, then factor congruence must
hold but it still doesn't prove that you can't get a
loop when C=1.
In fact, if you try to simply say
"there are no non-trivial attractors in 3n+1"
then the conjecture is false. The attractor -17C
found in every 3n+C system differs from the +C,
-C and -5C attractors in that it is NOT a trivial
attractor. And more importantly, it is NOT dependent
on factor congruence like all the other non-trivial
attractors. This means you can NEVER prove that there
are no non-trivial attractors in 3n+1 because we
have the counter-example in -17.
It may be true that there are no positive non-trivial
attractors but keep in mind that the seperate domains
of positive and negative integers is only a feature
of 3n+1 and doesn't apply to 3n+C.
And as far as Sequence Vecotrs / Hailstone Functions /
and Crossover Points are concerned, there is only one
domain, the integers (both positive and negative).
So basically, I've just proved the Collatz Conjecture
is false, so we can all go home now.
>
> I admit I have a lot to catch up on.
don't understand something, please reply to the
article so I can follow up. It is hard for me to
know when I'm being too obscure because I already
know the material.
Ernst
Do you have an interest in symbolism?
mensa...@aol.com
Ernst wrote:
> I ponder things.
>
> Do you have an interest in symbolism?
If you mean things like Kabala and numerology,
then yes, I have an interest in that I find that
stuff fascinating. Do I think there is any mystical
significance to it? No, because when you are free
to make up your own rules, you can do just about
anything.
For example, using the simple rule a=1, b=2, c=3, etc.
and ignoring punctuation,
ernstberg = 5 18 14 19 20 2 5 18 7
Added up, the sum is 108, which, when divided by 3 is 36.
The sum of integers from 1 to 36 equals 666.
So you must be an agent of the Anti-Christ.
When the numbers don't cooperate, I'll just make up rules
until they do. Take me,
mensanator = 13 5 14 19 1 14 1 20 15 18
These add up to 120, so I can't make the 666 trick work.
Not so fast, there are 10 letters in "mensanator", 120
divided by 10 is 12. Subtract 12 from 120 and you get 108,
same as your number.
So I must be an agent also.
Sometimes you have to be really creative to make things
work (wherein lies my fascination):
scimath = 19 3 9 13 1 20 8
Now group and sum the numbers
primes: 19 + 3 + 13 = 35
composites: 9 + 20 + 8 = 37
units: 1 = 1
Note that the primes plus units equals 36 and that the
composites minus the units also equals 36.
Notice how much this resembles the Collatz Conjecture
where the stopping point is 666. To make the symbolism
work, I just need a path to 36, any multiple of it, any
sum to it, any product to it, etc. The Beast Conjecture
is that for any word(s), a path exists to 666. If true,
then the mystical significance of your name summing to
108 evaporates.
If you showed someone the Collatz path from 27 to 1,
you could end by saying "Spooky, eh?". But if they
come to realize that all numbers lead to 1, they'll
say "So what, big deal.". Now it's interesting that
27 takes 111 steps to get to 1 and it's interesting
that "ernstberg" sums to 108, but that's all it is,
merely interesting, nothing more.
Ernst
Sometimes I wonder what the collatz models in the real world?
As to symbolism I like the binary realm of the Tao and the ilk.
I sometimes think A(x)+/-y,x/2 is more phenomenon than logic.
I mean if I decide to continue x/2 to infinity at the attractor then a
interaction between the two forces a(x)+/-y and x/2, part when the
sequence hits attractor with factors of 2.
well it's after work for me.. My program exited correctly and my
faith in powers of three as the Y or C is still intact.
I still see that there is no way to cheat Entropy. <sigh>
I am interested in in data compression as well.
I'm with you with your symbolism interests. i have not given those
areas much of my time but it's all good imo.
Ernst
I am reading what you have written...
The simple attractor program was just a [3x+y,x/2] where the last N
values are buffered and when a current value matches one in the buffer
then the loop is on.. The lowest value is then kept and the search
ends when that inital match value is seen again.
I wrote a hack program last time. This time I will use GMP.
Nothing to it really. I posted this info on my outdated web page long
ago... Man my web pages are a wreck.
mensa...@aol.com
Ernst wrote:
> BTW
>
> I am reading what you have written...
>
> The simple attractor program was just a [3x+y,x/2] where the last N
> values are buffered and when a current value matches one in the buffer
> then the loop is on.. The lowest value is then kept and the search
> ends when that inital match value is seen again.
That's what I do in Python. I add each number to a dictionary but
check first to see if the dictionary already contains that value,
if so, we've hit the loop.
A potential problem exists if you only keep the last N values.
There is no limit on how large a loop can be. If the loop happens
to be larger than the value of N you've chosen, you won't recognize
the loop when you encounter it (of course, in such a case the
symptom will be a failure of your program to terminate). My program
puts no limit on the buffer, I store EVERY number until either I find
the loop or I run out of memory.
Although I haven't run out of memory yet (and you probably haven't
found a loop that exceeds N), I can prove that somewhere there is
an attractor whose loop length exceeds the memory capacity of your
computer.
One of Python's advantages (although slower than compiled c code)
is it has more high level stuff than c. If your buffer is a simple
array, it will be inefficient for searching for a dulpicate number
in the sequence. Python's dictionary data structure uses a binary
tree and all that is done automatically. In c, you have to write
your own binary tree algorithm. I just picked up O'Reilly's
"Practical c" programming book. They have an example of how to
do a binary tree in c and when I get some free time, I'm going
to try and do my loop detector that way (with GMP, of course).
If I finish it, I'll post it over in your c programming thread.
> I wrote a hack program last time. This time I will use GMP.
> Nothing to it really. I posted this info on my outdated web page long
> ago... Man my web pages are a wreck.
Yeah, I bumped into them over the weekend. You had a link to your
attractor c program, but it was not found. Your list of attractors
is still there, though.
Ernst
If you don't mind what is an example size of a large loop? Mind ya, I
have hand coded my programs so by nature I didn't have 3 million binary
digit numbers or more as a test case before.
Also I ponder if your "test' number is a mersenne prime?
I think I read you have an interest in mersenne numbers
Yeah: I worked on compressing random data after sci.math.
i was able to reduce all binary integers by one bit; however, the
length of the string became an external requirement. It is a codec but
perhaps not a practical one yet.
We work in a fixed word size world, most of the time but, I assume
nature doesn't.
So my web pages fell to ruin due to my wonderlust. I can be like
that; from fire to fire.
Anyway:
I have never worked with tree's. I'm a simple programmer but I do
have Sedgewicks (sp?) Algorithms in C somewhere around here. Could
save us some time. It has all the tree's and other structures already
in it. I'll find it this weekend.
I should learn about trees and such.
Other than that; I have 1 gig of memory in this workstation and am
ready to add another. I just need a reason to spend the money. The
memory for this computer is still expensive.
Alright.. I have poker game here in a few. I'll be on the new posts
Saturday or Sunday.
I'll start the works over there. Are you are alright developing under
GPL ?
cool. I found being alone hour after hour, year after year, at the
keyboard to be boring and lonesome. It is good to share yet,
development details might be better shared in private before being made
public.
mensa...@aol.com
Ernst wrote:
> So you have seen large loops. I have only worked with smaller numbers.
>
> If you don't mind what is an example size of a large loop?
3n+1271069 has an attractor at 97 whose loop sequence has
315529 numbers (115819 odd, 199710 even). One of the things
I discovered was that for 3n+Y (where Y is not a power of 3),
the number 1 always leads to a non-trivial attractor. So I
did a test searching through various Y always starting at 1
and seeing how big the attractor's loop sequence was.
>From there I noticed that most of the record setting loop
lengths ocurred when Y was prime. Switching the test to use
GMP's next_prime function, I was able to locate the above.
> Mind ya, I have hand coded my programs so by nature I
> didn't have 3 million binary digit numbers or more as a
> test case before.
I don't think the numbers in that loop are all that large.
They're bigger than a standard c 32-bit long, so you probably
still need GMP (unless you c compile can do 64-bit longs).
Doing my Hailstone Function is another matter. For a loop that
size the parameters are something like 50000 digits. Luckily,
I don't need to use the Hailstone Function to find the attractor.
> Also I ponder if your "test' number is a mersenne prime?
I just checked and although 1271069 is prime, it is not a
Mersenne number.
> I think I read you have an interest in mersenne numbers
Yes, they are significant in 3n+1 as starting values because
they produce sequences longer than average, although usually
not record setters. I haven't investigated them in 3n+Y nor
have I specifically looked at when Y is a Mersenne Number.
>
>
> Yeah: I worked on compressing random data after sci.math.
> i was able to reduce all binary integers by one bit; however, the
> length of the string became an external requirement. It is a codec but
> perhaps not a practical one yet.
> We work in a fixed word size world, most of the time but, I assume
> nature doesn't.
> So my web pages fell to ruin due to my wonderlust. I can be like
> that; from fire to fire.
> Anyway:
> I have never worked with tree's. I'm a simple programmer but I do
> have Sedgewicks (sp?) Algorithms in C somewhere around here. Could
> save us some time. It has all the tree's and other structures already
> in it. I'll find it this weekend.
> I should learn about trees and such.
>
> Other than that; I have 1 gig of memory in this workstation and am
> ready to add another. I just need a reason to spend the money. The
> memory for this computer is still expensive.
>
> Alright.. I have poker game here in a few. I'll be on the new posts
> Saturday or Sunday.
>
> I'll start the works over there. Are you are alright developing under
> GPL ?
Sure, I don't care. I'm here for the love of the game.
>
> cool. I found being alone hour after hour, year after year, at the
> keyboard to be boring and lonesome. It is good to share yet,
> development details might be better shared in private before being made
> public.
Well, nobody seems to be active in our groups, so we're effectively
private.
Eric
One question and one remark.
1) -17c
in a previous post in this thread you said that -17c was not a trivial
attractor in any 3x+c system. Why ?
My understading is that -17c being an attractor in any 3x+c (c odd)
system is a direct and trivial consequence of -17 being an attractor
for the 3x+1 system,what have i missed ?
2) Mersenne number
I have to disagree with your statement that mersenne primes produce
longer sequences than average.
It's esay to determine and prove the theoretical average sequence
lenght (total stopping time) of any number x by ln(x)*2/ln(4/3)
Which gives n*2*ln(2)/ln(4/3) = 4.818*n for number in the 2^n-1 form.
That is very consistent with your own numerical calculations (on our
site).
What is funny in an other hand is that these number obviously have
sequence beginning with n odd steps leading to 3^n-1, we could have
expected them to have an higher average sequence length of n +
n*ln(3)/ln(4/3)*2.
It seems that regarding their behavior under collatz rules, numbers in
the form 2^n-1 are surprizingly more "normal" than numbers in the form
3^n-1
mensa...@aol.com
Eric wrote:
> Hi mensanator,
>
> One question and one remark.
>
> 1) -17c
>
> in a previous post in this thread you said that -17c was not a trivial
> attractor in any 3x+c system. Why ?
> My understading is that -17c being an attractor in any 3x+c (c odd)
> system is a direct and trivial consequence of -17 being an attractor
> for the 3x+1 system,
Yes, for any 3x+c system (including c=1) there are attractors at
+c, -c, -5c and -17c.
> what have i missed ?
I haven't made the [FAQ] post for the Crossover Point Function yet
where that is explained.
I changed the definition of "trivial" (sorry about that). I used to
say that those four attractors were the trivial attractors since
they all occur in 3x+1.
But I see now that that is not quite right. From my Crossover Point
function
Z * C
CP = -------
X - Y
CP is an integer (and thus, an attractor) trivially if X-Y is
either 1 or -1.
Sequence Vector: [2]
4*a - 1*c
Hailstone Function: g = ---------
3
1 * c c
Crossover Point: ----- = ----- = c
4 - 3 1
Sequence Vector: [1]
2*a - 1*c
Hailstone Function: g = ---------
3
1 * c c
Crossover Point: ----- = ----- = -c
2 - 3 -1
Sequence Vector: [1,2]
8*a - 5*c
Hailstone Function: g = ---------
9
5 * c 5 * c
Crossover Point: ----- = ----- = -5c
8 - 9 -1
So +c, -c and -5c are trivial attractors. But for -17c, we have
Sequence Vector: [1,1,1,2,1,1,4]
2048*a - 2363*c
Hailstone Function: g = ---------------
2187
2363 * c 2363 * c
Crossover Point: --------- = --------- = -17c
2048-2187 -139
For the Sequence Vector that produces -17c, the denominator
of the Crossover Point function is NOT 1 or -1 and yet we
have an integer result anyway because Z=17*139.
This is significant (and not trivial) because we cannot prove
that Z alone can never cancel all the factors of X-Y since there
is clearly a case where it does (albeit in the negative domain).
So all that Factor Congruence stuff doesn't prove there isn't
an attractor somewhere that does not depend on the factors of c.
>
> 2) Mersenne number
>
> I have to disagree with your statement that mersenne primes produce
> longer sequences than average.
But that depends on what you mean by "average". And also what is
meant by "length". If length means the count of odd numbers in the
sequence, the relation for Mersenne numbers -based on bit size-
is 4.818*bitsize.
If you look at the length for all n-bit numbers, the mean will be
about half that. That's what I meant by "longer than average".
> It's esay to determine and prove the theoretical average sequence
> lenght (total stopping time) of any number x by ln(x)*2/ln(4/3)
> Which gives n*2*ln(2)/ln(4/3) = 4.818*n for number in the 2^n-1 form.
> That is very consistent with your own numerical calculations (on our
> site).
My numbers were based on the statistics of propagating carry bits
extending the length of the binary number at the MSB end
while simultaneously removing bits from the LSB end.
> What is funny in an other hand is that these number obviously have
> sequence beginning with n odd steps leading to 3^n-1, we could have
> expected them to have an higher average sequence length of n +
> n*ln(3)/ln(4/3)*2.
When, for large n, 3^n-1 is converted to binary, the pattern of 1s
and 0s is a negative binomial distribution and is thus doomed to
decay rapidly. The record setters for bit length n diverge to their
excursion point more slowly but also decay more slowly once past it.
So although the n-bit Mersenne number is well above the mean n-bit
number, it is generally not an outlier in the distribution.
> It seems that regarding their behavior under collatz rules, numbers in
> the form 2^n-1 are surprizingly more "normal" than numbers in the form
> 3^n-1
I wouldn't say that. 3^n-1 numbers should be found near the mean
of the distribution for equivalent bit-length numbers, so you can't
be more "normal" than that. All 2^n-1 numbers become 3^n-1 numbers,
but with 1.585*n binary digits, so the fall from the excursion point
is normal for a 1.585*n-bit number but greater than that of an n-bit
number. And when you add in the time to reach excursion, you end up
about about twice the mean.