Let a = {(x, y) | y = 2x-1, X ∈ n *}, B = {(x, y) | y = AX2 ax + A, X ∈ n *}, ask whether there is a non-zero integer a, so that a ∩ B ≠? If it exists, request the value of a; if it does not exist, explain the reason

Let a = {(x, y) | y = 2x-1, X ∈ n *}, B = {(x, y) | y = AX2 ax + A, X ∈ n *}, ask whether there is a non-zero integer a, so that a ∩ B ≠? If it exists, request the value of a; if it does not exist, explain the reason

Suppose a ∩ B ≠ 0, then the system of equations y = 2x − 1y = AX2 − ax + A has positive integer solution. By eliminating y, we get AX2 - (a + 2) x + A + 1 = 0. (*) from △≥ 0, we get (a + 2) 2-4a (a + 1) ≥ 0, and the solution is - 233 ≤ a ≤ 233. Because a is a non-zero integer, ∩ a = ± 1, when a = - 1, we substitute (*), and the solution is x = 0 or x = - 1, and X ∈ n *. So a ≠ - 1. When a = 1, we substitute (*), and the solution is x = 1 or x = 2 Therefore, there exists a = 1 such that a ∩ B ≠ a ∩ B = {(1,1), (2,3)}