[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?
Thanks,
Tony
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