Originally I want to prove $y^{p'} \equiv x^n + C \pmod p$ is always having integer solution for some prime $p$ and $p'$
It is given by my classmate, so I do not know if it can really be proved, but this question is about my approach as stated below:
If $y \equiv x^n + C \pmod 3 $ has integer solution for some integers $n$ and $C$,
then is it correct to say, by Chinese Remainder Theorem:
there exists some integer $z$ which $y \equiv z^n + C \pmod p$ for some prime $p$?
As $3$ and $p$ must be co-prime.
If yes, then I think the original statement is also proved (assuming I can prove $y \equiv x^n + C \pmod 3$)
If no, please tell me what is wrong behind my reasoning of this approach. Thanks.