Mathematics

Problem

Let

S = \{\,(n,\, n^{2} - 2n) \mid n \in \mathbb Z\,\}

. a. (i) Show that every point of the set

S

lies on the curve

y = x^{2} - 2x

. b. (ii) Determine all integers

n

for which the corresponding point in

S

also lies on the parabola

y^{2} = 4x

and is located in the first quadrant. c. (iii) List explicitly the points that satisfy the conditions obtained in part (ii).