I have an equation in $x$ and I would like to determine if it has any solutions modulo a large prime $p$. Suppose $p$ is large enough that I can factor numbers up to $p$, but I cannot test all values up to $p$. (Actually, so far, I have been doing just that -- but I'd like to avoid this as it takes a long time. If you can avoid factoring, all the better.)
The particular equation I have is $$ x^4-x^2\equiv4\pmod p $$ but I would be interested in
- Solutions to this particular problem, or more generally
- Solutions to other quadratics$\pmod p$ in $x^2$, or more generally
- Solutions to quartics$\pmod p$.
I'm familiar with quadratic reciprocity but not with cubic or biquadratic. (It's not clear to me if this can be transformed so they can be used; if so, demonstrating the transformation and giving a pointer to a good source on higher reciprocity would suffice as an answer.)