Date: Jan 4, 2013 9:53 PM
Author: David Bernier
Subject: Re: Another count sort that certainly must exist, it do not have<br> any restrictions upon size of (S number of possibilities)

On 01/04/2013 08:55 PM, JT wrote:
> On 5 Jan, 02:39, David Bernier<david...@videotron.ca> wrote:
>> On 01/04/2013 12:07 PM, David Bernier wrote:> On 01/04/2013 10:46 AM, JT wrote:
>>>> On 4 Jan, 15:46, JT<jonas.thornv...@gmail.com> wrote:
>>>>> I remember doing this in a tentamen during my education in information
>>>>> theory beleiving what i did was binary sort but my teacher informed me
>>>>> it wasn't so what is it.

>>
>> [...]
>>
>>
>>
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>>
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>>

>>>> heap, what is the difference betwee a heap and a tree?).
>>
>>>> So what you think about the mix using this kind of sort for counting
>>>> in values, and then quicksort to sort the none null tree nodes by
>>>> sizes.

>>
>>> Oops.. below is about factoring. The best algorithms
>>> have been getting better since Maurice Kraitchik's [1920s]
>>> improvement on Fermat's method of expressing a number
>>> as a difference of squares, n = a^2 - b^2, so
>>> n = (a-b) (a+b).

>>
>>> There's a very good article called "A Tale of Two Sieves"
>>> by Carl Pomerance: Notices of the AMS, vol. 43, no. 12,
>>> December 1996:
>>> <http://www.ams.org/notices/199612/index.html>

>>
>>> The 9th Fermat number F_9 = 2^(512)+1 had been factored
>>> around 1990 by the Lenstras et al using the Number Field
>>> sieve (which had supplanted the quadratic sieve).

>>
>>> The Quadratic sieve is easier to understand than the
>>> Number Field Sieve, which I don't understand.

>>
>>> F_10 and F_11 were fully factored then, using the elliptic
>>> curve method (which can find smallish prime factors).

>>
>>> F_12 was listed as not completely factored, with
>>> F_12 being a product of 5 distinct odd primes and
>>> the 1187-digit composite:

>>
>>> C_1187 =
>>> 22964766349327374158394934836882729742175302138572\

>>
>> [...]
>>

>>> 66912966168394403107609922082657201649660373439896\
>>> 3042158832323677881589363722322001921.

>>
>>> At 3942 bits for C_1187 above, what's the
>>> probability density function of expected time
>>> till C_1187 is fully factored?

>>
>> For the Fermat number F_12 = 2^(2^12) + 1 or
>> 2^4096 +1 , another prime factor was found around
>> 2010. So, this new prime factor would be a divisor
>> of C_1187, a 1187-digit number. F_12 is listed
>> as known to be "not completely factored".
>>
>> The relevant line on the Web-page referenced below contains
>> the text: "M. Vang, Zimmermann& Kruppa" in the "Discoverer"
>> column:
>> <http://www.prothsearch.net/fermat.html#Complete>
>>
>> Also, lower down in the page,
>> "50 digit k = 17353230210429594579133099699123162989482444520899"
>>
>> This does relate to a factor of F_12 by PARI/gp.
>> Then, by my calcultions, the residual unfactored part
>> of F_12 has 1133 decimal digits and is a composite number.
>>

>>> Or, centiles: e.g. 50% chance fully factored
>>> within<= 10 years. 95% chance fully factored within
>>> <= 95 years, etc. ...

>>
>> Maybe 50% to 50% chances for "fully factored by 2100 " ?
>> (or 2060, or 2200 etc. ... )
>>
>> dave

>
> Is reading out a binary tree from smallest to biggest in anyway
> related to TSP, i have slight memory of doing a algoritmic solution
> for TSP when i tried to factor RSA primeproducts, unfortunatly it is
> all gone from my mind. Could you please make a short layman approach
> for how TSP and sorting is related to factorization it may come back
> to my memory. I lack the key so to speech to the problem.


The key lies within the mind. The mind may appear to be a part
of reality, but in reality, all is mind ...

dave