Problem 59 **

Screenshot 2020-07-07 at 20.42.47

x^2+3yx-2y^2=0
4x^2+12xy+9y^2-17y^2=488
(2x+3y)^2-17y^2=488
(2x+3y)^2(\mod17)-0(\mod17)=12(\mod17)

We just need to show that k^2\neq 12(\mod17) for any integer k.

This is quite easy to do just by checking the remainders upon division by 17 for 0^2, 1^2,2^2,...,16^2, and you’ll find that the only remainders are 0,1,2,4,8,9,13,15,16.