Re: Q:The Mersenne Primes and the Collatz conjecture (3x+1)/2.
- From: "mensanator@xxxxxxxxxxx" <mensanator@xxxxxxx>
- Date: 9 Dec 2005 15:12:47 -0800
Dan wrote:
> Observing once again the following path in the Collatz
> tree.
>
> .655360,327680,163840,81920,40960,20480,10240,5120,
> 2560,1280,640,320,160,80,40,20,10,5,16,8,4,2,1
>
> Stating with (10) this is a point of entry into this
> sequence with Mersenne (1) 2^2-1 = 3.
>
> From here on out the points of entry just before
> entering this sequence for all Mersenne primes
> (have)? to be a prime??.
First of all, not every MP enters that sequence.
>
> This would make 10,40,160,not 640,2560,10240,not 40960,
> not 163840,655360,..etc
>
> The entry points are 10,40,160,2560,10240,655360,..
> and NO entry points @ 640,40960,163840,.. because
> (640-1)/3 = a composite and (40960-1)/3 = a composite
> also (163840-1)/3 = a composite.
>
> Is this true or false?
Is every third sub-branch of a branch composite? Yes.
Sub-branches always follow mod 3 sequence:
n+2 mod 3____x*32
|
x*16
|
n+1 mod 3____x*8
|
x*4
|
n mod 3____x*2
|
x
So one of every 3 sub-braches of ANY branch is divisible
by 3 and thus composite.
>
> This is taken from early observations of the Mersenne
> primes journey back to ..16,8,4,2,1 so I could be
> wrong here!
That is vacuously true, ALL sequences end ..16,8,4,2,1.
Now if you meant ..5,16,8,4,2,1 then it's false.
MP(21) and MP(24) both have paths ending
...341,1024,512,256,128,64,32,16,8,4,2,1
>
> If I am wrong, please give a counter example of an odd
> composite*3 +1 that enters this sequence starting with
> some larger Mersenne prime.
341 = 11 * 31
But then, that's not the sequence you're talking about.
On the other hand, 341 IS a sub-branch of the powers of 2
branch as is 5. So if we define "this sequence" to mean
any Order 1 branch{1}, then the answer is no, 341 being
a composite stem of an Order 1 branch through which at
least two Mersenne Primes pass.
{1} Order is the level of sub-branches relative to the
powers of 2 branch (Order 0). Thus, 5, 21, 85, 341
are all Order 1 branches as they are 1 (3x+1)
iteration away from the Order 0 branch. 3 would be
an Order 2 branch since it's 2 (3x+1) iterations
away from the Order 0 branch.
>
> Thanks,
>
> Dan
.
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