Re: The Collatz discrete primes!

On Apr 2, 9:53 am, "Danny" <fasttrac...@xxxxxxxxxxxxx> wrote:
On 1 Apr, 11:35, "mensana...@xxxxxxxxxxx" <mensana...@xxxxxxx> wrote:





On Mar 31, 9:43 am, "Danny" <fasttrac...@xxxxxxxxxxxxx> wrote:

WTF? Google is trying to censor my work!

Trying agin with some of the detail trimmed.

It appears you do not want to commit to the idea that
an odd pure (only) will trigger a counter example
if indeed a counter example exists?

I don't want to commit without proof, no.

I could be wrong here in my assumption.

And you could be right. What you CAN do is take the assumption
and see if either true or false leads to a contradiction.

Explaining it better, I still maintain that if there is a
counter example and you are doing a sequential seed
search,

This is important - IF you are doing a sequential seed search.

which is the only way to find all the pures,

You haven't proved that sequential seed search is the ONLY
way to find all pures.

For example, take the 1(mod3) odd integer

34802787818142911388658951020295077885411469701175231988550894382577351419

there's no way purity can resolved by sequential seed searching.
But all I have to do to demonstrate impurity is to show a smaller
number that leads to it in aCollatzsequence:

3562191662859645825772777219239755072232830437249231770406464115509311
10686574988578937477318331657719265216698491311747695311219392346527934
.
.
.
<sequence trimmed of 483 large numbers>
.
.
.
556844605090286582218543216324721246166583515218803711816814310121237622704
278422302545143291109271608162360623083291757609401855908407155060618811352
139211151272571645554635804081180311541645878804700927954203577530309405676
69605575636285822777317902040590155770822939402350463977101788765154702838
34802787818142911388658951020295077885411469701175231988550894382577351419

And the seed number

3562191662859645825772777219239755072232830437249231770406464115509311

is indeed smaller than the hailstone

34802787818142911388658951020295077885411469701175231988550894382577351419

so the hailstone is impure. I don't know whether I can find pure
numbers with similar techniques, but I haven't given it any thought.

then the first odd integer would have to be pure
that leads into or is part of this loop.

And I don't dispute that.

With a further explanation, if it were not a pure
odd then how would you explain that every integer
connected with this counter example has to be pure
to the entireCollatztree.

I'm unsure what you mean here. "Entire" I take to mean
the union of the disjoint trees. "Trivial" means the tree
rooted at 1. "Non-trivial" would be a tree rooted at the
counter example. If you mean every odd integer on the
non-trivial tree is pure with respect to the trivial tree
(vacuous), ok. But not all integers on the non-trivial
tree are absolutely pure, they are impure with respect to
other numbers on the non-trivial tree. This is what I meant
earlier, does the definition of pure have to take into
account disjoint trees?

So as a counter example it would have to be an infinite
pure tree made infinite because of the reverse doubling
effect where this pure tree falls outside of the infinite
Collatztree!

A counter example tree cannot be infinitely pure. Here I'm
saying all the odd integers could be vacuously pure with
respect to the trivial tree while also being impure with
respect to the counter example tree. There cannot be a tree
where every odd integer is pure with respect to both trees.

In the above sentance translation ---
(infinite = ---->oo) or approching infinity.

Hard to imagine because now when this pure is found
in the sequential seed search,IF EVER, it creates more
pures in theCollatztree but these pures will never
be entered in the sequential seed search for pures in the
Collatztree.

Why wouldn't they? Every number is part of the sequential
seed search.

In other words skipped over because they are not part of
theCollatztree.

No, they wouldn't be skipped, what would happen is that they
would never be encountered in any priorCollatzsequence which
implies they are pure. But the definition of "pure" you gave
also implies thatCollatzis true (has only the trivial tree).
IfCollatzis false, this implication is actually "vacuously
pure" and you will have to further determine whether it is
absolutely pure (pure on the non-trivial tree).

Only the first key seed
odd pure will be entered exposing the counter example.

Then who is to say that another or more counter examples
exist creating 2 or more sepparate trees outside theCollatz
tree?

Of course, if there is one counter example, there's no reason
to think there couldn't be many. That's the signicance of the
-17 counter example. Not because it's in the negative domain,
but it proves that the set of counter examples can't be empty,
so there cannot be a proof that the set of counter examples is
empty.

Is this just a little food for thought or my logic gone
awry in the pursuit of a proof of the 3x+1 problem?

Like a lot of things, it reamins food for thought until it's
utilized to make a proof. Personally, I don't see how purity
can help, but then again, I don't have a proof, so who cares
what I think.

Dan

You make valid points but it is hard for me to come around
to your way of thinking on pure or impure as being the smallest
odd integer triggering a counter example. So your thought is
if there is a counter example it could be a pure or an impure odd
that triggers the counter example.

I'm not saying it could be pure or impure. I'm saying that I don't
know whether it must be one or the other. "Being pure" and "must
be pure" are two different things. The first case simply means that
the counter example just happens to be pure. The second case means
that a counter example cannot exist unless it's pure. I believe your
conjecture is the second case. I'm saying I don't think you've
proved that and even a confirming example doesn't prove it since
it could be the first case purely by coincidence.


The trivial tree with all its pure trivial 0(mod 3)'s and the more
seleted complex 1(mod 3)'s is one thing but any thing outside the
trivial tree (counter example tree) is also pure odd or even in
its relation to the trivial tree.

Right, but what about the counter example tree itself? What do you
call numbers that are pure relative to the trivial tree that are
impure relative to the counter example tree? And what do you call
a number that's pure relative to both trees? I've been using
"vacuously pure" for a number that's on the counter example tree and
pure to the trivial tree and "absolutely pure" if pure relative
to both. I haven't thought of a term for a counter example number
pure-relative-to-trivial that's impure-relative-to-counter-example.


As far as generating these pures in the trivial tree the 0(mod 3)'s
are trivial because it involves all of them but the more complex
1(mod 3)'s where only some are pure is the hitch in generating
a complete list of pures without a sequential seed search.

But I don't know that a sequential seed search is necessary to find
pures. It might be, but I don't know that for a fact and won't
assume it. It's definitely not necessary to find impures (and here
I'm ONLY talking about 1(mod3) since the others are already known).
I also don't know whether I can find ALL impures. If I could, then
whatever's left must be pure.


Sure you can find many impures by checking for any 2(mod 3)'s,
all impure, but find a very large 1(mod 3) that is pure without
a sequential seed search.

At the present, this is unknown and I'm not ruling it out.

Or even a large 1(mod 3) that is impure
like in your example through trial and error but that is impure.

Ah, but it wasn't trial and error, it was a closed form function
and that was the point, an example of finding intractably large
(with respect to sequential seed searching) impures to demonstrate
that some purity questions can be answered without sequential seed
searching. In particular, it was a 26th Generation Type [1,1,1,1,1,4]
Sequence Vector Hailstone. ALL type [1,1,1,1,1,4] hailstones are
impure as I noted in one of my earlier posts.

I don't know if there is a closed form way of identifying pure
numbers.


As it stands now there is no known closed form to find these
elusive 1(mod 3)'s thus a sequential seed search is necessary.

There is also no proof that it can't be done. And the fact that I
can find impures via a closed form certainly makes one think a
closed form for pures might be possible.


A good analogy is like the factorials, there is no closed
form method to fine any given factorial but only a gamma function
approximation.

A better analogy is the Multi-Generation Hailstone problem.

Generally, this requires a recursive function to solve:

def gen0_recursive(k,xyz):
"""Find first member of generation k of Hailstone Function.

gen0(k,xyz)
k: generation
xyz: HailstoneFunctionParameters
Locates the first _a_ for the requested generation k of
the Hailstone Function.
Needs to know _d_ and _c_ from previous generation, so it
calls itself recursively until generation 1.
At generation 1, results are simple linear congruence of x,y,z.
a: the first solution to the Hailstone Function
g: seed that genertes _a_
d: difference between _a_ and _g_
j: index of _a_ where d == 0 (mod y**k)
c: a[j] the
returns [0,0,0,0] if k invalid
returns GenerationParameters [a,g,d,c]
"""
if k<1: return [0,0,0,0]
if k==1:
a = gmpy.divm(xyz[2],xyz[0],xyz[1])
g = seed(a,xyz)
d = a - g
c = a
return [a,g,d,c]
else:
prev_gen = gen0(k-1,xyz)
j = gmpy.divm(xyz[1]**(k-1)-prev_gen[2], \
xyz[1]-xyz[0], \
xyz[1]**(k-1))/xyz[1]**(k-2)
a = j*xyz[1]**(k-1) + prev_gen[3]
g = seed(a,xyz)
c = a
d = a - g
return [a,g,d,c]

xyz = [8,9,5]
for g in xrange(1,250):
print '%4d ' % cf.geni(1,g,xyz),

4 13 22 31 40 49 58 67 76 85
94 103 112 121 130 139 148 157 166 175
184 193 202 211 220 229 238 247 256 265
274 283 292 301 310 319 328 337 346 355
364 373 382 391 400 409 418 427 436 445
454 463 472 481 490 499 508 517 526 535
544 553 562 571 580 589 598 607 616 625
634 643 652 661 670 679 688 697 706 715
724 733 742 751 760 769 778 787 796 805
814 823 832 841 850 859 868 877 886 895
904 913 922 931 940 949 958 967 976 985
994 1003 1012 1021 1030 1039 1048 1057 1066 1075
1084 1093 1102 1111 1120 1129 1138 1147 1156 1165
1174 1183 1192 1201 1210 1219 1228 1237 1246 1255
1264 1273 1282 1291 1300 1309 1318 1327 1336 1345
1354 1363 1372 1381 1390 1399 1408 1417 1426 1435
1444 1453 1462 1471 1480 1489 1498 1507 1516 1525
1534 1543 1552 1561 1570 1579 1588 1597 1606 1615
1624 1633 1642 1651 1660 1669 1678 1687 1696 1705
1714 1723 1732 1741 1750 1759 1768 1777 1786 1795
1804 1813 1822 1831 1840 1849 1858 1867 1876 1885
1894 1903 1912 1921 1930 1939 1948 1957 1966 1975
1984 1993 2002 2011 2020 2029 2038 2047 2056 2065
2074 2083 2092 2101 2110 2119 2128 2137 2146 2155
2164 2173 2182 2191 2200 2209 2218 2227 2236

Note that this list contains two Mersenne Numbers, 31 & 2047.

Turns out ALL the Mersenne Numbers from that list can be found
using a closed form expression, even though the general solutions
can't be. How bizarre is that?

def Type12MH(k,i):
"""Find ith, kth Generation Type [1,2] Mersenne Hailstone
using the closed form equation

Type12MH(k,i)
k: generation
i: member of generation
returns Hailstone (a)
"""
ONE = gmpy.mpz(1)
TWO = gmpy.mpz(2)
SIX = gmpy.mpz(6)
NIN = gmpy.mpz(9)
if (k<1) or (i<1): return 0
i = gmpy.mpz(i)
k = gmpy.mpz(k)
a = (i-ONE)*NIN**(k-ONE) + (NIN**(k-ONE) - ONE)/TWO + ONE
return TWO**(SIX*a - ONE) - ONE

for g in xrange(1,8):
p = cf.Type12MH(1,g)
print p,

31 2047 131071 8388607 536870911 34359738367 2199023255551

So, if you can't show me a proof that there is no closed form way
to find pures, I'm not ruling it out.

In this problem we need more than an approximation, it needs to be
exact!

I use Python, so everything I do is exact. I can generate an infinite
number of odd 1(mod3) impures without using sequential seed
searching.
But keep in mind that this is not the same as proving a particular
number is impure.

You could be right, that sequential seed searching is the only
way to find a pure. But I will never just assume that without reason.


Dan

.

Relevant Pages

  • Re: 3n+5 pures and impures!
    ... You stated that they be impure. ... to be pure wheather it be the smallest odd integer, ... Any of the numbers in the counter example tree? ... Crossover Point of the sequence vector ...
    (sci.math)
  • Re: 3n+5 pures and impures!
    ... You stated that they be impure. ... to be pure wheather it be the smallest odd integer, ... the trivial tree, impure sequence vectors have to be connected ... Crossover Point of the sequence vector ...
    (sci.math)
  • Re: 3n+5 pures and impures!
    ... You stated that they be impure. ... to be pure wheather it be the smallest odd integer, ... the trivial tree, impure sequence vectors have to be connected ... Crossover Point of the sequence vector ...
    (sci.math)
  • Re: 3n+5 pures and impures!
    ... You stated that they be impure. ... to be pure wheather it be the smallest odd integer, ... Any of the numbers in the counter example tree? ... Crossover Point of the sequence vector has ...
    (sci.math)
  • Re: 3n+5 pures and impures!
    ... You stated that they be impure. ... to be pure wheather it be the smallest odd integer, ... Any of the numbers in the counter example tree? ... Crossover Point of the sequence vector has ...
    (sci.math)