Consider a quadratic equation with integer coefficients.
Is every integer a possible discriminant?

Prove it.

(In reply to Four by Bractals)

Also, an odd integer not congruent to 1 mod 4 cannot be the discriminant of a quadratic with integer coefficients.

If bē-4ac=d and all letters stand for integers, then d is congruent to bē mod 4. But the square of an even number is congruent to 0 mod 4 and the square of an odd number is congruent to 1 mod 4. Hence d must be congruent either to 0 or to 1 mod 4.

On the other hand, given an integer d congruent either to 0 or to 1 mod 4, it is easy to  find integers a,b, and c such that bē-4ac=d.

Thus an integer is the discriminant of a quadratic with integer coefficients if and only if it is congruent either to 0 or to 1 mod4.

Posted by Richard on 2005-05-25 09:10:41