Consider the following attempt to prove this true theorem:

*Theorem:* $a^3+b^3=c^3$ has no solutions with $a$, $b$ and $c$ positive integers.

*Attempted proof:* Suppose that there are positive integers $a$, $b$ and $c$ such that $a^3+b^3=c^3$.

I We have $a^3=c^3-b^3$.

II Hence $a^3=(c-b)(c^2+cb+b^2)$.

III It follows that $a=c-b$ and $a^2=c^2+cb+b^2$, since $a\le a^2$ and $c-b\le c^2+cb+b^2$.

IV Eliminating $a$, we have $(c-b{)}^2=c^2+cb+b^2$.

V Multiplying out, we have $c^2-2cb+b^2=c^2+cb+b^2$.

VI Hence $3cb=0$ so one of $b$ and $c$ is zero.

But this is a contradiction to the original assumption that all of $a$, $b$ and $c$ are positive. It follows that the equation has no solutions.

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