Suppose we have two positive integers $a$ and $b$ which satisfy the condition $a^3 − 2b^3 = 2$.

Question

Suppose we have two positive integers $a$ and $b$ which satisfy the condition $a^3 − 2b^3 = 2$.

It then follows that the greatest common divisor of $a$ and $b$ must be either $1$ or $2$. True or false?

I tried solving by supposing $a$ and $b$ are composite numbers sharing $\beta$ as a common factor.

Letting $a = \alpha \beta$ and $b = \gamma \beta$, I factorised $\beta^3 (\alpha^3 + 2 \gamma^3) = 2$, $$\beta^3 = \frac{2}{\alpha^3 + 2 \gamma^3},$$ which is lesser than a whole number therefore both $a$ and $b$ must be coprime to each other, i.e $\gcd(a, b) = 1$ only.

Answer 1

Let be $d=gcd(a,b)$ so $a=d.a_1$ and $b=d.b_1$, with $gcd(a_1,b_1)=1$ and backing to the equation we have: $$a^3-2b^3=2 \Rightarrow d^3(a_1^3-2b_1^3)=2 \Rightarrow d^3|2 \Rightarrow d=1.$$