Fundamental Discriminant

In mathematics, a fundamental discriminant D is an integer invariant in the theory of integral binary quadratic forms. If Q(x, y) = ax2 + bxy + cy2 is a quadratic form with integer coefficients, then D = b2 − 4ac is the discriminant of Q(x, y). Conversely, every integer D with D ≡ 0, 1 (mod 4) is the discriminant of some binary quadratic form with integer coefficients. Thus, all such integers are referred to as discriminants in this theory. Every discriminant may be written as

D = D₀f 2

with D₀ a discriminant and f a positive integer. A discriminant D is called a fundamental discriminant if f = 1 in every such decomposition. Conversely, every discriminant D ≠ 0 can be written uniquely as D₀f 2 where D₀ is a fundamental discriminant. Thus, fundamental discriminants play a similar role for discriminants as prime numbers do for all integers.

There are explicit congruence conditions that give the set of fundamental discriminants. Specifically, D is a fundamental discriminant if, and only if, one of the following statements holds

• D ≡ 1 (mod 4) and is square-free,
• D = 4m, where m ≡ 2 or 3 (mod 4) and m is square-free.

The first ten positive fundamental discriminants are:

1, 5, 8, 12, 13, 17, 21, 24, 28, 29, 33 (sequence A003658 in OEIS).

The first ten negative fundamental discriminants are:

−3, −4, −7, −8, −11, −15, −19, −20, −23, −24, −31 (sequence A003657 in OEIS).