Let $B$ be a positive integer. Show that the torsion subgroup of the elliptic curve $y^2 = x^3 + B$ has order dividing $6$ (Feel free to assume Dirichlet's theorem, that for any coprime $a$ and $d$ there are infinitely many primes $p$ with $p\equiv a \pmod d$, but not any stronger version of it.)
The discriminant is $\Delta = 27B^2$ so we can use reduction mod $p$ for any prime $p\geq 5$, $p\nmid B$. Using primitive roots one can easily show that if $p\equiv 2 \pmod 3$ then every remainder mod $p$ is a cube and hence that $y^2 = x^3 + B$ has exactly $p+1$ points modulo $p$ (including the point at infinity). So the order of the torsion group must divide $p+1$ for all $p\equiv 2 \pmod 3$. Then what?
Any help appreciated!