FKMO 1997 Problem 6

Let $p_1,p_2,\dots,p_r$ be distinct primes, and let $n_1,n_2,\dots,n_r$ be arbitrary natural numbers. Prove that the number of pairs of integers $(x, y)$ such that $$x^3 + y^3 = p_1^{n_1}p_2^{n_2}\cdots p_r^{n_r}$$ does not exceed $2^{r+1}$ .