<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] $$a,b,c=\text{odd}\quad\iff\quad a=2A+1\quad;\quad b=2B+1\quad;\quad c=2C+1$$ $$\Delta=b^2-4ac=n^2\quad\iff\quad(2B+1)^2-4\,(2A+1)(2C+1)=n^2$$ $$(4B^2+4B+1)-4\,(4AC+2A+2C+1)=n^2$$ $$4\,\Big[(B^2+B)-(4AC+2A+2C+1)\Big]+1=n^2$$ $$\iff n=2k+1\iff n^2=4k^2+4k+1\iff$$ $$\underbrace{\underbrace{B(B+1)}_{even}-\underbrace{2\,(2AC+A+C)}_{even}-1}_{odd}=\underbrace{k(k+1)}_{even}$$ Contradiction !
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