Okay, I have this for a class homework question. If we are given n and p, I know a few things:
- If p≡1 (mod 3), then we can use the theorem of cubic reciprocity to determine whether there exists an integer x such that it falls into n's congruence class.
- If p≡2 (mod 3), we can say that there definitely is an integer x such that it falls into n's congruence class.
- If p≡0 (mod 3), I don't know.
The issue is that when using this theorem the best we can do is say that there exists an x, not identify the original x. The question asks whether we can ever know what x originally was, given its remainder mod p.
I don't know if this is more properly under CS stackexchange, but this felt much more mathematical in nature.