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Topic: Re: New "good" examples of Hall's conjecture.
Replies: 0

 Ismael Jimenez Calvo Posts: 1 Registered: 11/28/13
Re: New "good" examples of Hall's conjecture.
Posted: Nov 28, 2013 4:42 PM

Resent-From: <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: "sci-math-research@moderators.isc.org"
<sci-math-research@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!

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. 2435-2444 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
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