Show the following equivalence: X2 + DY 2 properly represents 2m+2 ?? 4X2 +2BXY + CY2 (D + B2 =4C)..

Show the following equivalence: X2 + DY 2 properly represents 2m+2 ?? 4X2 +2BXY + CY2 (D + B2 =4C) properly represents 2m+2 by (x,y), with y =0 mod 2. [ Sketch of proof. First note that AX2 +2XY +CY2 = 1 A[(AX +BY)2 + DY 2]. Thus 4X2 +2BXY +CY2 =2m+2 ?? (2X +BY 2 )2 +D(Y 2 )2 =2m+2. Then, you have to ?nd the argument for the correctness of the transformations ? =2x + By 2 x = 1 2(?-B?)? = y 2 y =2 ? ]