How to find the number of solutions to $ 2x^{4}+x^{3}-4x+11\equiv 0 \pmod{1000000}$?

Question

I know that for a quadratic equation we can look at if $ b^2-4ac $ is a quadratic residue and conclude that there are two solutions but I am not sure how to do this with a quartic equation. Is there a way I could apply the method above or is there another way?

Answer 1

It is easy to check by hand that $1$ is the unique solution modulo $2$ and modulo $5$. You could then use the nice method proposed by Mark Bennet to solve the problem, but is not so easy to find the solutions predicted by Hensel's lemma.

It is actually a bit depressing to find out that a brute force approach using the Sage software instantaneously gives the unique solution, x = 717071 .

for x in [0..99999]:
  x = 10*x + 1;
  if (((2*x^4 + x^3 - 4*x + 11) % 1000000) == 0):
    print "x = " + str(x)

x = 717071