<s>[INST] Prove that if $a, b, c$ are positive odd integers, then $b^2 - 4ac$ cannot be a perfect square. Prove that if $a, b, c$ are positive odd integers, then $b^2 - 4ac$ cannot be a perfect square.
What I have done:
This has to either be done with contradiction or contraposition, I was thinking contradiction more likely.
 [/INST] Suppose $b^2-4ac$ is a square. Since $a,b,c$ are odd, $b^2-4ac$ must be an odd square. Hence $x=\frac{-b+\sqrt{b^2-4ac}}{2}$ is an integer. Since $x^2+bx+ac=0$, therefore $x^2+bx+ac\equiv 0\,(mod\,2)$. By FLT $x^2\equiv  x\,(mod\,2)$. Hence $(1+b)x+ac\equiv 0\,(mod\,2)$. Since $2|1+b$. Thus $0+ac\equiv 0\,(mod\,2)$ (contradiction)
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