[seqfan] Determine whether a number is represented by a binary quadratic form

T. D. Noe noe at sspectra.com
Sun Jul 11 02:08:38 CEST 2010

The are many sequences generated by binary quadratic forms.  Suppose we
have a quadratic form

	f(x,y) = ax^2 + bxy + cy^2

with integers a, b, and c.  For integer N, we want to know whether N =
f(x,y) has a solution for x and y relatively prime.  It appears that we can
look at the reduced equation

	az^2 + bz + c = 0 (mod N)

If it has an integer solution z=z0, then there is a solution to N = f(x,y).
In fact, the number of solutions to the reduced equation is the same as the
original equation

Is there a name for this process?  Can the solution (x,y) be determined
from z0?



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