If x2 is a sum of two consecutive positive integers y and z, then x2+y2=z2. Consider, x2+y2=(y+z)+y2=(y+y+1)+y2=y2+2y+1=(y+1)2=z2. Equations of the form x2−dy2=1,...If x2 is a sum of two consecutive positive integers y and z, then x2+y2=z2. Consider, x2+y2=(y+z)+y2=(y+y+1)+y2=y2+2y+1=(y+1)2=z2. Equations of the form x2−dy2=1, where x,y∈Z, and d is a positive integer which is not a square of an integer. Prove that if x=a and y=b is a solution, then x=a2+2b2 and y=2ab is also a solution.
