To what sets must $a,b,c$ belong?

Question

I just thought of kind of a cool number theory/algebra problem.

Given that $$\sqrt{b^2-4ac}\in\Bbb N$$ To which sets must $a,b,c$ belong?

It is obvious that $$b^2-4ac\in\left\{x^2|x\in\Bbb N\right\}$$ But beyond that, I do not know what to do. May I have some help?

Edit:

To make things more interesting, what if $a,b,c\in\Bbb N$?

Note that $a,b,c$ need not be integers. So this is no Diophantine equation. So it is a little bit "less cool". We just write $b^2-4ac=x^2$, so that $a=(b^2-d^2)/(4c)$ for $c$ nonzero. Then choose $b,c\in \Bbb R$ and $d \in \Bbb Z$. — Dietrich Burde, Mar 10 '19 at 20:21

Will Jagy · Answer 1 · 2019-03-10T21:50:37.810

3

all triples $a,b,c$ that work are given by $$ ax^2 + b x y + c y^2 = (sx+ty)(ux+vy) \; , \; $$ $$ a = su, $$ $$ b = sv+tu, $$ $$ c = tv . $$ When those happen, $$ b^2 - 4ac = s^2 v^2 + 2 stuv + t^2 u^2 - 4 stuv =s^2 v^2 - 2 stuv + t^2 u^2 = (sv-tu)^2 $$

see Prove that if $b^2-4ac=k^2$ then $ax^2+bx+c$ is factorizable

edited Mar 10 '19 at 21:50

answered Mar 10 '19 at 21:45

Will Jagy

139,541

Okay, so to which sets do $s,v,t,u$ belong? – clathratus Mar 11 '19 at 01:11
@clathratus integers – Will Jagy Mar 11 '19 at 01:18

To what sets must $a,b,c$ belong?

1 Answers1