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Re: Another count sort that certainly must exist, it do not have any restrictions upon size of (S number of possibilities)
Posted:
Jan 4, 2013 8:39 PM


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 = (ab) (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 1187digit 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 1187digit number. F_12 is listed as known to be "not completely factored".
The relevant line on the Webpage 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



