Re: [Collatz] was : Re: Status of Waring-problem - Collatz - Sorry

From: Mensanator (mensanator_at_aol.compost)
Date: 08/29/04 

Date: 29 Aug 2004 19:19:13 GMT

>Subject: Re: [Collatz] was : Re: Status of Waring-problem - Collatz - Sorry
>From: Gottfried Helms helms@uni-kassel.de
>Date: 8/28/04 4:56 PM Central Daylight Time
>Message-id: <cgqvaa$2k8$01$1@news.t-online.com>
>
>Am 28.08.04 18:47 schrieb Mensanator:
>[big snip, being a bit tired at this evening]
>>
>> That's interesting. One of the things I was looking at is why Mersenne
>> numbers, which have the highest excursion (largest value in sequence)
>> don't ever seem to be sequence length record holders. Part of this seems
>> to be related to the glide. The slope of the line up to the excursion is
>steep,
>> but so is the fall from the excursion. The record holders have a gentler
>slope
>> up to a lesser excursion, but also a gentler slope down from the excursion
>> producing a longer glide. I haven't had much luck constucting sequence
>length
>> record holders, so maybe the secret is focusing on the glide.
>>
>>
>Mersenne numbers are that of the form 2^N - 1 , right?

Yes.

>
>If you look at series of trajectories of some transformation of
>the same length, say
> writetrajectory(findfirst(A,B,C,D))
>modifying the exponents A,B,C,D, then there are two important
>observations:
>
>
>==> for i:=1 to 8 do
> writeln({i,1,1},":", trajectory({i,1,1}));
>end;
>.
>Exponents: ! lowest trajectory
>T(a;A,B,C) !
>-----------!------------------
>(1, 1, 1) :(15, 23, 35, 53)
>(2, 1, 1) :(9, 7, 11, 17)
>(3, 1, 1) :(61, 23, 35, 53)
>(4, 1, 1) :(37, 7, 11, 17)
>(5, 1, 1) :(245, 23, 35, 53)
>(6, 1, 1) :(149, 7, 11, 17)
>(7, 1, 1) :(981, 23, 35, 53)
>(8, 1, 1) :(597, 7, 11, 17)
>
>here we see, that the outcoming is only 53 or 17 (53=2*3^3 -1 and
>17=2*3^2-1)
>depending on the remainder of the first exponent modulo 2. Note, that only
>the trajectory for T(a;1,1,1) requires a first member "a" of the
>mersenne-form
>2*2^3-1 = 15.
>
>Now we change the second exponent:
>
>==> for i:=1 to 8 do
> writeln({1,i,1},":", trajectory({1,i,1}));
>end;
>.
>Exponents: ! lowest trajectory
>T(a;A,B,C) !
>-----------!------------------
>(1, 1, 1) :(15, 23, 35, 53)
>(1, 2, 1) :(27, 41, 31, 47)
>(1, 3, 1) :(19, 29, 11, 17)
>(1, 4, 1) :(67, 101, 19, 29)
>(1, 5, 1) :(163, 245, 23, 35)
>(1, 6, 1) :(99, 149, 7, 11)
>-----------------------------------------
>(1, 7, 1) :(995, 1493, 35, 53) * repeating modulo class 53
>(1, 8, 1) :(1763, 2645, 31, 47)
>
>Now we have six different results as endpoint of the complete
>trajectory, depending on the remainder of the second exponent
>modulo 6. Also we can see, that they all are below 54
>
>Now changing 3'rd exponent:
>
>==> for i:=1 to 21 do
> writeln({1,1,i},":", trajectory({1,1,i}));
>end;
>.
>Exponents: ! lowest trajectory
>T(a;A,B,C) !
>-----------!------------------
>(1, 1, 1):(15, 23, 35, 53)
>(1, 1, 2):(7, 11, 17, 13)
>(1, 1, 3):(55, 83, 125, 47)
>(1, 1, 4):(87, 131, 197, 37)
>(1, 1, 5):(23, 35, 53, 5)
>(1, 1, 6):(407, 611, 917, 43)
>(1, 1, 7):(663, 995, 1493, 35)
>(1, 1, 8):(1175, 1763, 2645, 31)
>[...]
>(1, 1, 18):(970903, 1456355, 2184533, 25)
>-----------------------------------------
>(1, 1, 19):(4116631, 6174947, 9262421, 53) * repeating modulo class 53
>(1, 1, 20):(2019479, 3029219, 4543829, 13)
>(1, 1, 21):(14602391, 21903587, 32855381, 47)
>
>we get all residue classes 6k+1, 6k-1 of 54, which
>is 2*3^3 and gives 18 residue-classes that way.
>The important observation is, that the mersenne-requiring
>combination of exponents (always T(a;1,1,1,1,...1) )
>also gives the highes value of the residue-classes.
>
>These "mersenne"-trajectories are the same as I call
>them "primitve transformation" (or "primitve loop" if a
>descendent step is added) and transform from
>
> 2*2^(N-1) - 1 ----> 2*3^(N-1) - 1
>
>where the rhs is the highest value in the residue class of
> 2*3^(N-1)
>
>*All* other transformations T(a;A,B,C) land in a lower class
>than T(a;1,1,1). One could state that as a theorem.
>Now we have lots of other combinations of exponents of the
>same length; some are keeping the following members constantly
>above the starting value:
>
> T(a;1,2,1) for instance gives
>
>Exponents: ! lowest trajectory
>T(a;A,B,C) !
>-----------!------------------
> (1, 2, 1):(27, 41, 31, 47)
>
>So it is easy to construct trajectories, which do not fall
>below their initial value, and this need not explicitely
>be proven.
>
>Since T(a;1,1,1) lands on the highest residue-class of 2*3^3
>all other transformations of the same length land on lower
>classes, but since we can always construct trajectories, which
>as well do not fall below their initial value, we can conclude, that
>we always can construct a glide of the same length as the mersenne-
>trajectory but with a lower ending point.
>
>(hope I didn't miss your point, possibly it's getting too late)

Thanks for taking the time to do that.
I'm going to have to study your web site some more,
this is great stuff.

>
>good night -
>
>Gottfried Helms 

-- 
Mensanator
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