All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Numbers
Discriminants (Posted on 2005-05-24) Difficulty: 2 of 5
Consider a quadratic equation with integer coefficients.
Is every integer a possible discriminant?

Prove it.

See The Solution Submitted by Jer    
Rating: 3.5000 (4 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
re: Four | Comment 4 of 7 |
(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

Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (11)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2024 by Animus Pactum Consulting. All rights reserved. Privacy Information