Terry M
Posts:
15
Registered:
4/26/12
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Re: Pell type equations
Posted:
Apr 28, 2012 2:21 PM
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"Helmut Richter" <hhr-m@web.de> wrote in message
news:alpine.LNX.2.00.1204281339210.6033@badwlrz-clhri01.ws.lrz.de...
> 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
Aren't the quadratic residues of 8 only 1 and 2?
I thought for the quadratic residue
x^2 = a (mod n)
a and n have to be coprime.
> 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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