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Topic: Pell type equations
Replies: 29   Last Post: Apr 29, 2012 5:37 PM

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Helmut Richter

Posts: 164
Registered: 7/4/06
Re: Pell type equations
Posted: Apr 28, 2012 10:58 AM
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On Sat, 28 Apr 2012, Terry M wrote:

> "Helmut Richter" <hhr-m@web.de> wrote in message
> news:alpine.LNX.2.00.1204280933150.5441@badwlrz-clhri01.ws.lrz.de...
>

> > Whenever you see a Diophantine equation of the form
> >
> > ax² + by² + c = 0
> >
> > you should, before thinking, see what happens if taken modulo m for each m
> > which is:
> >
> > - the number 8
> >

>
> I think I understand the following two, but why the number 8 ?


Just because 8 has so few quadratic residues (0, 1, and 4) that you have a
chance that ax² + by² cannot get all values, with good luck not the value
of -c, e.g. 3x² + 7y² is never 6 (mod 8). A test modulo 4 would not have
sufficed.

Needless to say that passing all tests does not mean that there are
solutions. An example is x² + 378y² + 6 = 0 with no solutions (from an old
posting of mine <slrnc5ld51.rgt.a282244@lxhri01.lrz.lrz-muenchen.de>; I
did not double-check now).

--
Helmut Richter



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