> "Helmut Richter" <firstname.lastname@example.org> wrote in message > news:alpine.LNX.email@example.com... > > > 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 <firstname.lastname@example.org>; I did not double-check now).