Problem

Source: 67 Polish MO 2016 Second Round - Problem 4

Tags: number theory, set, Perfect Squares, Poland



Let $k$ be a positive integer. Show that exists positive integer $n$, such that sets $A = \{ 1^2, 2^2, 3^3, ...\}$ and $B = \{1^2 + n, 2^2 + n, 3^2 + n, ... \}$ have exactly $k$ common elements.