Terry M
Posts:
15
Registered:
4/26/12


Re: Pell type equations
Posted:
Apr 28, 2012 2:21 PM


"Helmut Richter" <hhrm@web.de> wrote in message news:alpine.LNX.2.00.1204281339210.6033@badwlrzclhri01.ws.lrz.de... > On Sat, 28 Apr 2012, Terry M wrote: > >> "Helmut Richter" <hhrm@web.de> wrote in message >> news:alpine.LNX.2.00.1204280933150.5441@badwlrzclhri01.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.lrzmuenchen.de>; I > did not doublecheck now). > >  > Helmut Richter

