5

Quadratic forms

**5.1 Equivalence**

We shall consider binary quadratic forms

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

where *a*, *b*, *c* are integers. By the discriminant of *f* we mean the number *d* = *b*^{2} − 4*ac*. Plainly *d* ≡ 0 (mod 4) if *b* is even and *d* ≡ 1 (mod 4) if *b* is odd. The forms *x*^{2} − *dy*^{2} for *d* ≡ 0 (mod 4) and *x*^{2} + *xy* + (1 − *d*)*y*^{2} for *d* ≡ 1 (mod 4) are called the principal forms with discriminant *d*. We have

4af (x, y) = (2ax + by)^{2} − dy^{2}, |

whence if *d* < 0 the values taken by *f* are all of the same sign (or zero); *f* is called positive or negative definite ...

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