The support problem and its elliptic analogue (Q1358801)

Suppose that \(x,\;y\) are in any algebraic number field \(F\) and have the following property: \(y^n \equiv 1 \pmod{\mathbf p}\) whenever \(x^n \equiv 1 \pmod {\mathbf p}\) for almost all prime ideals \({\mathbf p}\) of \(F\) and all \(n \in {\mathbb Z}\). The authors then prove that \(y\) is a power of \(x\). This result solves a question of Pal Erdős (1988). An analogue for elliptic curves \(E\) over \(F\) is also given: If \(P,Q \) are \(F\)-rational points of \(E\) and \(nQ=0\) whenever \(nP=0\) (in \(E(\mathbb{F}_P)\)), then \(Q=fP\) for some endomorphism \(f\) of \(E\) (or \(P,Q\) are torsion points).