The Diophantine equation $x^2 - 97y - 40 = 0$

Question

I am trying to determine whether the equation below has a solution or not

$$x^2-97y-40 =0.$$

If a solution exists, $x^2-40$ must be congruent to $0$ modulo $97$.

If I could show the congruence above implies that solution exists.

Thanks for your help...

Answer 1

you can use legendre symbol for calculating quadratic residue.so you need to compute (40/97).

  (40/97)=(2/97).(2/97).(2/97).(5/97)

        =(2/97).(5/97)

        =(97/2).(97/5)

        =(1/2).(2/5)

       =1.(-1)=-1

so 40 is not a quadratic residue mod 97.

no solution exists.