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A Comprehensive Course in Number Theory by Alan Baker

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5

Quadratic forms

5.1 Equivalence

We shall consider binary quadratic forms

f (x, y) = ax2 + bxy + cy2,

where a, b, c are integers. By the discriminant of f we mean the number d = b2 − 4ac. Plainly d ≡ 0 (mod 4) if b is even and d ≡ 1 (mod 4) if b is odd. The forms x2imagedy2 for d ≡ 0 (mod 4) and x2 + xy + image(1 − d)y2 for d ≡ 1 (mod 4) are called the principal forms with discriminant d. We have

4af (x, y) = (2ax + by)2dy2,

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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