Quadratic Mathematics | Part IB, 2002

Explain what is meant by saying that a positive definite integral quadratic form f(x,y)=ax2+bxy+cy2f(x, y)=a x^{2}+b x y+c y^{2} is reduced, and show that every positive definite form is equivalent to a reduced form

State a criterion for a prime number pp to be representable by some form of discriminant dd, and deduce that pp is representable by a form of discriminant 32-32 if and only if p1,2p \equiv 1,2 or 3(mod8)3(\bmod 8). Find the reduced forms of discriminant 32-32, and hence or otherwise show that a prime pp is representable by the form 3x2+2xy+3y23 x^{2}+2 x y+3 y^{2} if and only if p3(mod8)p \equiv 3(\bmod 8).

[Standard results on when 1-1 and 2 are squares (mod pp ) may be assumed.]

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