We say that a point P = (x, y) in R 2 is rational if both x and y are rational. More precisely, P is rational if P = (x, y) Q 2 .

We say that a point P = (x, y) in R 2 is rational if both x and y are rational. More precisely, P is rational if P = (x, y) ∈ Q 2 . An equation F(x, y) = 0 is said to have a rational point if there exists x0, y0 ∈ Q such that F(x0, y0) = 0. For example, the curve x^2 + y^2 −1 = 0 has rational point (x0, y0) = (1,0). Show that the curve x^2 + y^2 −3 = 0 has no rational points.
Exercise 20 (above) involved showing that there are no rational points on the curve x^2 + y^2 −3 = 0. Use this fact to show that p 3 is irrational.
Explain why x^2 + y^2 −3 = 0 not having any rational solutions (Exercise 20) implies x^2 + y^2 −3k = 0 has no rational solutions for k an odd, positive integer (In problem 2 interpret Explain as Prove. That is, prove x^2+y^2−3k=0 has no rational solutions when k is an odd integer.)