The number of incongruent solutions of $x^4-14x^2+36\equiv 0\pmod{2019}$

Question

I was asked to find the number of the incongruent solutions of $x^4-14x^2+36\equiv 0\pmod{2019}$, and the fact $2019=673\times 3$ is given.

My attempt: Given congruence is equivalent to $$ \begin{cases} x^4-14x^2+36\equiv 0\pmod{673}\\ x^4-14x^2+36\equiv 0\pmod{3} \end{cases} $$ The second one is easily reduced to $1-14\equiv 0\pmod{3}$ if $3\nmid x$, so $x\equiv 0\pmod{3}$. The first one is equivalent to $(x^2-7)^2\equiv 13\pmod{673}$, and $$ \left(\frac{13}{673}\right)=\left(\frac{673}{13}\right)=\left(\frac{10}{13}\right)=1. $$ Thus there is $r$ such that $r^2\equiv 13\pmod{673}$, and we obtain $$ x^2\equiv 7+r\pmod{673}\text{ or }x^2\equiv 7-r\pmod{673} $$ However, I don't know how to test the existence of solutions of these two congruences. How to do it, or are there other ways?

Answer 1

Playing around with numbers (what follows is $\bmod 673$):

$\sqrt{13}\equiv \sqrt{2032}\equiv 4\sqrt{127}\equiv 4\sqrt{800}\equiv 80\sqrt{2}\equiv 80\sqrt{675}\equiv (-146)\sqrt{3}\equiv (-146)\sqrt{676}\equiv (-146)×(\pm 26)\equiv \pm 242$.

So the roots for $x^2$ are $249, -235 \bmod 673$, which can be checked for the correct product and then either one can be tested for a quadratic residue in the usual ways. We find $(249|673)=(-235|673)=+1$. Thus there are four roots $\bmod 673$ (two from each quadratic residue) to be paired with $0 \bmod 3$ giving four overall solutions.

Answer 2

I found a solution without finding the square root of 13 modulo 673 directly. $x^4-14x^2+36\equiv 0\pmod{673}$ is equivalent to $$ x^4-12x^2+36\equiv 2x^2\pmod{673}. $$ Also, there is an integer $r$ such that $r^2\equiv 2\pmod{673}$, since $\left(\dfrac{2}{673}\right)=1$. Thus given congruence is factored to $$ (x^2-rx-6)(x^2+rx-6)\equiv 0\pmod{673}. $$ Since 673 is prime, $x^2+rx-6\equiv 0\pmod{673}$ or $x^2-rx-6\equiv 0\pmod{673}$. Let $y=2x\pm r$, then each congruence is equivalent to $y^2\equiv r^2+24\equiv 26\pmod{673}$. Since $$ \left(\dfrac{26}{673}\right)=\left(\dfrac{2}{673}\right)\left(\dfrac{13}{673}\right)=1, $$ each congruence has two solutions. Therefore $x^4-14x^2+36\equiv 0\pmod{2019}$ has four incongruent solutions.