"Helmut Richter" <email@example.com> wrote in message news:alpine.LNX.firstname.lastname@example.org... > On Sat, 28 Apr 2012, Terry M wrote: > >> "Helmut Richter" <email@example.com> wrote in message >> news:alpine.LNX.firstname.lastname@example.org... >> >> > 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 <email@example.com>; I > did not double-check now). > > -- > Helmut Richter