[seqfan] Re: Determine whether a number is represented by a binary quadratic form
Georgi Guninski
guninski at guninski.com
Sun Jul 11 09:39:29 CEST 2010
On Sat, Jul 10, 2010 at 05:08:38PM -0700, T. D. Noe wrote:
> 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?
>
looks like if b = 0 the equation must have solutions mod a and mod c, e.g.:
2 x^2+2^3 y^2=13 [1]
mod 13 == n:
2x^2+2^3y^2 = 0 <=> z^2+2^2 = 0, z = x/y = 3 is a solution
the LHS of [1] is even and RHS is odd.
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