I need help solving a problem for my discrete class.

The problem is : Prove no integer solutions of x and y satisfy the equation . Hint: Consider this equation in modulo 5.

My attempt:

Assume

Thus, in mod 5,

Note that 5y^2 in mod 5 will always be 0, since it will be a multiple of 5, thus:

Now I broke it case by case to see what x^2 will have to be to satisfy the above requirement.

I found that

in order for

implies

The following steps may be useless and/or incorrectly derived, but i dont know what else to do:

if 5k+3 = x^2 , then

What now?