How would I solve the following set of modular equations where $p$ and $q$ are primes.
$N = p^3 (\mod q)$
$N = -q^3 (\mod p)$
If I know $N$, is it possible to solve and find $p$ and $q$ using Chinese Remainder Theorem? Chinese remainder theorem tell us how to solve this the other way around, but how can I solve it knowing N?
Not every $a$ in $\{0,1,\cdots,(q-1)\}$ is a cube mod $q$. If you know $N,$ there may be a way to find if it is a cube mod $q,$ however it's not as direct as e.g. quadratic reciprocity. A good possibility is to look up "cubic reciprocity" for which some facts are known.
Do the same for $N$ mod $p,$ and if say $N=e^3$ mod $p$ then $N=-(-e)^3$ mod $p.$ so it seems not much issue to "putting solutions together." '