```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.>>>> [...]>>>>>>>>>>>>>>>>>>>>>> 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 partof reality, but in reality, all is mind ...dave
```