Re: Primes and the Collatz conjecture!

>Subject: Re: Primes and the Collatz conjecture!
>From: [EMAIL PROTECTED]  (Mensanator)
>Date: 8/23/03 1:35 PM Central Daylight Time
>Message-id: <[EMAIL PROTECTED]>
>
>>Thanks for you're in depth reply. You have much more expertise then I
>>on any math involved here or any other math-related subject I am sure.
>>My only objective here was basically to show these special primes that
>>reside in this special sequence as the starting  (seed) integer for
>>this one path in the Collatz tree. Where each node off this path is
>>only one and that is where all of these special odd primes and 0 (mod)
>>3's and other odd composites with prime factors reside thus creating
>>only 3 odd integers, the (seed), prime 5, and 1 in their entire path
>>starting with (seed) 3. I am just picking the primes related by one
>>node to this one path. Also there is no 0 (mod) 5 in any of these odd
>>composites (seeds).
>>
>>As in the Mersenne primes, not all-prime exponents of 2 create another
>>prime. The same is true here, but with a twist, not all-even exponents
>>(n) where (2^n*10-1)/3 will create another prime. You also have to
>>consider the *10 factor of the Collatz (seed) primes when comparing
>>densities of the two.
>>
>>Sorry if I misled anyone here into thinking there was some kind of
>>pattern, but I never had that intention.
>>
>>I am doing a comparison table of the Mersenne primes and the Collatz
>>(seed) primes. Yes, there is many more even exponents creating the
>>Collatz (seed) prime then the Mersenne prime exponents, so that will
>>explain the density factor between the two.
>>
>>A question also remains, is they're a Mersenne prime or Mersenne
>>primes member(s) of this path in the Collatz tree?
>
>When using tree structures to represent Collatz sequences, the ORDER
>is the number of branches a number is from the trunk (the branch whose
>root is 1). The ORDER is simply the number of odd integers (or the count
>of 3x+1 iterations) in the sequence, where branch 1 is considered ORDER 0.
>
>In your list, all your primes are sub-branches of branch 5 .  Each new prime 
>on your list attaches higher and higher up on branch 5 requiring more and 
>more itertions of x/2, but they all are ORDER 2.
>
>Now a Mersenne number (2^n - 1), whether prime or not, never has an
>ORDER less than n. Typically, the ORDER is n*4.819. That formula is not
>exact, although the error gets proportionally smaller as n increases. See 
>
>http://members.aol.com/mensanator666/Page.htm
>
>(In that chart, "Cycles" is used as a synonym for ORDER)
>
>So 3, which is 2^2 - 1, is the only possible Mersenne Number that can 
>(and does) appear on your list.

Also, as I mentioned before, every 2nd through nth sub-branch appends a 01 onto
the binary pattern of the first sub-branch:

11
1101
110101
11010101
.
.
.

Since Mersenne numbers are always all 1s in binary

1
11
111
1111
11111
.
.
.

if a Mersenne number appears as a sub-branch, it must be the first one and
there can be no further ones attached to that branch.

>
>Other Mersenne numbers will have sequences that pass through 5 on their
>way to 1. 31 passes through the prime 53, but it is ORDER 39 (and along
>with 27) is one of the only two numbers whose ORDER is larger than itself.
>
>>   
>>Thanks for your interesting input.
>>
>>Dan
>
>
>
>--
>Mensanator
>2 of Clubs   http://members.aol.com/mensanator666/2ofclubs/2ofclubs.htm


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