Representations by Quadratic Forms — Allen Hatcher
Mathematics — Number Theory · quadratic-reciprocity-rational-points
Problem
This exercise involves using quadratic reciprocity to apply Legendre's Theorem (Theorem 2.6) on rational points on quadratic curves.
(a) Determine the values of n for which the curve 2x2+ny2=1 contains rational points, assuming n is odd and squarefree. For each of the first three positive values of n for which the curve contains rational points find two of these rational points that lie in the first quadrant.
(b) For the same equation show that the case that n is even and squarefree reduces to the case n is odd and squarefree.
(c) Determine the values of n for which the curve 3x2+ny2=1 contains rational points, assuming n is odd, squarefree, and coprime to 3.