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Re: New "good" examples of Hall's conjecture.
Posted:
Nov 28, 2013 4:42 PM


ResentFrom: <bergv@illinois.edu> From: Ismael Jimenez Calvo <ijcalvo@gmail.com> Subject: Re: New "good" examples of Hall's conjecture. Date: November 28, 2013 at 9:11:21 AM MST To: "scimathresearch@moderators.isc.org" <scimathresearch@moderators.isc.org>
El jueves, 3 de octubre de 2013 18:02:00 UTC+2, Hans Kristian Ruud escribiÛ: > I here present these 4 new "good" examples of Hall's conjecture, > > using the terminology of Calvo et al [1]: > x^3  y^2 = k, > k < \sqrt{x}, > r = \sqrt{x}/k > > Note that the last example exhibits the _lowest_ value of r (1.006) of > known examples. > > 1) > x 69586951610485633367491417 > y 580485956840055298243897234120309242222 > k 6855536488571 > r 1.22 > > 2) > x 39739590925054773507790363346813 > y 250515780595813121159570482265776894494367196625 > k 1682917776799172 > r 3.75 > > 3) > x 862611143810724763613366116643858 > y 25335098466671949144333881652710390910673829348417 > k 26655452290421177 > r 1.10 > > 4) > x 1062521751024771376590062279975859 > y 34634326278079440059697495108987559220558998514272 > k 32394686429925205 > r 1.006 > > These examples were found by employing an algorithm proposed by > professor emeritus StÂl Aanderaa at the mathematics department > of the University of Oslo. A first version of the program was written > in > Python (in order to explore the feasibility of the algorithm); > a second version was written in C using the GNU MP library. > > As well as finding the new "good examples" given above, > the program has also found all of the the examples 2 up to 42 in the > list of "good examples" given in [1]. > > The program also has found the solution reported at [2]: > > x 10747835083471081268825856 > y 35235585373909771093395629910293233497 > z 2428129973007 > r 1.35 > > > > The first entry in the list of Calvo et al is > > x=2, y=3, k=1 > > This particular example is not found by the algorithm because the value > x = 2 fails to fulfil certain conditions which are fulfilled for all > greater values of x. > > The program uses about 1628 hours (slightly less than 70 days) of CPU > time to find the items 1 > 42 of the list of Calvo et al. and the , > the > example 1) above. > > A more detailed exposition will be posted on the arXiv website. > > References: > > [1] Calvo, Herranz and Saez: > A NEW ALGORITHM TO SEARCH FOR SMALL NONZERO #"5ÿeÐ^3 " 5ÿfÐ^2#" > VALUES, > MATHEMATICS OF COMPUTATION > Volume 78, Number 268, October 2009, Pages 2435 2444 > > [2] The homepage of I. Jimenez Calvo: http://ijcalvo.galeon.com/hall.htm >
Congratulations!
I am eager to know about your new algorithm and read your forthcoming publication. Only your second example of the Marshal Hall conjecture for x = 3973 ... 6813 may be found by the algorithm published in Math. Comp. 78(2009), pp. 24352444 in a reasonable time, in particular for the parameters b = 41376077211, C= 4 of that algorithm. If you agree, I can include your new examples in the table in http://ijcalvo.galeon.com/hall.htm crediting properly your new findings.
I know that some efforts, without success, has been done to "cover" the "good examples o Hall's conjecture" by series alike to those due to Schinzell, Danilov and Elkies. Therefore, the search for "good examples of Hall's conjecture remains as a hard computational problem.
Ismael Jimenez Calvo



