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

prove that the diophantine equation x^ 27y^2 = 67 has no solutions using exhaustion of cases

67 is odd

So, both x and y cannot be odd or both x and y cannot be even

So we have two cases

Case 1: x is odd, y is even

x=2k+1,y=2m

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

So,

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

But, 67=3 mod 4

So no solutions possible in this case

Case 2: x is even , y is odd

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

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

So again no solutions

Hence no integer solutions possible