Proving there are no integer solutions for $3x^2=9+y^3$

Question

Prove there are no $x,y\in\mathbb{Z}$ such that $3x^2=9+y^3$.

Initial proof

Let us assume there are $x,y\in\mathbb{Z}$ that satisfy the equation, which can be rewritten as $$3(x^2-3)=y^3.$$ So, $$3 \mid y \Rightarrow 3^3 \mid y^3 \Rightarrow 3^2 \mid x^2 - 3.$$ As $3 \mid -3$, it follows that $$3 \mid x^2 \Rightarrow 3^2 \mid x^2.$$ Say $x^2=3^2 \cdot a^2$ for some $a \in \mathbb{Z}$. Then $$\begin{align*}x^2-3 &= 3^2 \cdot a^2 - 3 \\&= 3 \cdot (3 \cdot a^2 - 1)\end{align*}$$ As $3^2 \mid x^2-3$, it follows that $3 \mid 3 \cdot a^2 - 1$. It is obvious that $3 \mid 3 \cdot a^2$, so it follows that $3 \mid -1$, which is false. Therefore, the assumption that there are $x,y\in\mathbb{Z}$ was wrong.

Alternative proof

$$ \begin{align*} 3x^2&=9+y^3\\ 3(x^2-3)&=y^3 \end{align*} $$

So, $3 \mid y \Rightarrow 3^3 \mid y^3 \Rightarrow 3^2 \mid x^2 - 3 \Rightarrow 3^2 \mid x^2 \Rightarrow 3^2 \not\mid x^2 - 3$ and we are done.

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Both proofs are essentially the same, except for the fact that the second proof is much shorter. I think the first proof is much more clear, but it takes a bit longer to write down.

• Would the second proof suffice in a math competition, or is it too short?
• This problem was part of the Dutch finals of 1978. I figured it out in under a minute, which is quite unusual for me, so now I wonder: is my proof correct?

Answer 1

Your second proof starts without an assumption and this is not a matter of length. The word "Assume" before the first formula would suffice. This oversight would probably be forgiven if your work ended with a contradiction and the consequences which it does not.

Between $3^2\mid x^2-3$ and $3^2 \mid x^2$, you should add at least $3\mid x$ and ideally $3\mid x^2-3 \Rightarrow 3\mid x^2 \Rightarrow 3\mid x$.

You are highly likely to lose some points on that step, although it is not certain that you would.

The difference between you first and second version is NOT the shortness and lack of words, but the fact that you left out that step.

The second issue is that you do not say in your second proof that there are no integer solutions. You are certain to lose points on that. So, for example, you could circle the two contradictory assertions, make arrows to the next line and write "Contradiction. There are no solutions."

It is not an issue of length, but whether every step is addressed in some way.

Answer 2

Suppose there is a solution to $3x^2 = 9 + y^3$.

Then $3$ divides the RHS hence $3|y^3$, so $3|y$. Let $y = 3a$ for some integer $a$.

Then $3x^2 = 9 + 27a^3$. Hence $9 | 3x^2$ and so $3|x^2$, implying $3|x$. Let $x = 3b$ for some integer $b$.

Then $27b^2 = 9 + 27a^3$, and after cancelling $9$'s we get $3b^2 = 1 + 3a^2$ or $3(b^2 - a^2) = 1$, which clearly has no solutions since the LHS is divisible by $3$.