Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 29 / 2 / v / Solution

We can use reduction instead of expanding a very large rational addition formula. Modulo $2$, both $P_1$ and $P_2$ reduce to $(0,0)$, which generates the three-element reduced group. The reduction homomorphism therefore givesThe rational point $P_2$ is not any of the three torsion points just found, so it has infinite order. In particular $R=P_1+5P_2\ne O$: otherwise multiplication by three would give $15P_2=O$.

If $R$ had integral coordinates, its affine integral representative would reduce to an affine point modulo $2$, not to the point at infinity. This contradicts its reduction to $O$. ThusIndeed, it is not even integral at $2$. This is a reduction certificate for a nonintegral elliptic point, with the crucial nonidentity condition verified by the torsion calculation.