Date: Jan 4, 2013 8:39 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 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.

[...]

>> 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