[seqfan] Re: Finding numbers represented by indefinite binary quadratic forms

[Giovanni Resta]

On 6/4/2014 4:14 PM, Neil Sloane wrote:
> Dave, Thanks, that Mathematica command does indeed seem to do the job! It
> gives (as you say) a long list of solutions if solutions exist, and "false"
> if they don't.

Yes, in general the Reduce command applied to quadratic Diophantine 
equations gives False if no solutions are found, a list of solutions if
solutions are in finite number, or one or more solutions in exponential
form, something like
x == 1/24 (-4 + 2 ((5 - 2 Sqrt[6])^(2 C[1]) + (5 + 2 Sqrt[6])^(2 C[1])))
with C[1] integer, if there exist infinite solutions.

In case just a few equations have to be solved, there is an online 
applet by  Dario Alpern at this address:

http://www.alpertron.com.ar/QUAD.HTM

A nice thing about that page is that it provides also a step-by-step
solution and that infinite solutions are represented by recurrences
instead of exponential expressions.

Giovanni