prove that the diophantine equation x 27y2 67 has no soluti

prove that the diophantine equation x^ 27y^2 = 67 has no solutions

67 is odd so one of x,y must be even and other odd

Case 1: x odd, y even

Hence, x^2=1 mod 4,y^2=0 mod 4

x^2-7y^2=1-0=1 mod 4

BUt, 67=3 mod 4

So no solutions possible

Case 2:x is even , y is odd

x^2=0 mod 4

y^2=1 mod 4

x^2-7y^2=0-7*1=-7=1 mod 4

So again no solutions

Hence no integer solutions possible