Solve the equation $x^3 + 117y^3 = 5$ over the integers.

Question

Solve the equation $x^3 + 117y^3 = 5$ over the integers.

I have tried solving this. It is clear that one of $x$ or $y$ must be negative. $117$ seemed a strange number. So I found out that $117 = 125 - 8 = 5^3 - 2^3$. I don't know if this is useful but still I'm adding it. So the equation becomes:

$$x^3 + (5y)^3 - (2y)^3 = 5$$

I don't know how to proceed further. I need some hints. Any help would be appreciated.

Answer 1

Hint: $$x^3 \equiv 0,1,8 (\bmod 9)$$ $$117y^3 \equiv 0 (\bmod 9)$$ $$5 \equiv 5 (\bmod 9)$$