If I take the following elliptic formula over a finite field of size $17$:
$$y^2 = x^3 + 2x + 3$$
The solutions for $x = 2$ would be $7$ and $10$.
Because
$7^2=49$ and $49 \equiv 15 \bmod 17$
$10^2=100$ and $100 \equiv 15 \bmod 17$
My question is when I take a number larger than $17^2$ I will still only get the solutions $7$ and $10$. For instance $24^2 = 576$ and $576 \equiv 15 \bmod 17$ and $24 \equiv 7 \bmod 17$.
Does this go on until $\infty$... OR does anything larger than $17$ just not exist