Re: Q:The Mersenne Primes and the Collatz conjecture (3x+1)/2.


Dan wrote:
> 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.
>
> Where I was wrong also was I assumed that all primes
> entered this sequense ...160,80,40,20,10,5,16,8,4,2,1

A lot of them do, just not all:

Entry points (last odd number prior to a power of 2)
and count of the 1st 1,000,000 odd primes that sequence
through them.

5: 937383
85: 24011
341: 37994
5461: 86
21845: 496
349525: 27
1398101: 2
22369621: 1



>
> Thanks,
>
> Dan

.

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