The Support Problem and Its Elliptic Analogue

Abstract LetFbe a number field, Supposex, y∈F* have the property that for alln∈Zand almost all prime ideals p of the ring of integers ofF* one has thatyn≡1 (mod  p ) wheneverxn≡1 (mod  p ). We show that thenyis a power ofx. This answers a question of Erdős. We also prove an elliptic analogue of this result.