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

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

RHS is 67=1 mod 3

We know squares modulo 3 can take values 0 or 1

Case 1 x^2=0 mod 3

x^2-7y^2=-7y^2 mod 3

We need this to be equal to 67 hence 1 mod 3

So,

-7y^2 =1 mod 3

-y^2=1 mod 3

y^2=-1 mod 3

Case 2 x^2=1 mod 3

x^2-7y^2=1-7y^2=1 mod 3

-7y^2=0 mod 3

Hence y=0 mod 3

So we see there is a solution

Now note that RHS is 67 which is odd.

And LHS is difference of 2 terms one of which must be odd and other must be even for RHS to be odd

Case 1 x= odd

Hence y must be even

So, x^2=1 mod 8

y^2=0 ,4 mod 8

x^2-7y^2=1,-3 mod 8

Case 2. y is odd

y^2=1 mod 8

x=0,4

x^2-7y^2=0-7 or 4-7 mod 8

x^2-7y^2=-7,-3 mod 8

But 67=3=-5 mod 8

Hence no solutions