```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 or2^4096 +1 , another prime factor was found around2010.  So, this new prime factor would be a divisorof C_1187, a 1187-digit number. F_12 is listedas known to be "not completely factored".The relevant line on the Web-page referenced below containsthe 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 partof 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
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