If $n + x$ with $n, x \in \Bbb N$ is prime, is it possible to prove generally, that $n^m + x$ with $n, x, m \in \Bbb N$ is not prime for at least one $m$? If yes, how can this be done?
EDIT: There was one comment stating that this is not possible, if you take $n = 1$ and for example $x = 2$. That is right. I am interested in solutions for $n > 1$.
Let $$ P = n+x $$ be prime, as stated. What can you say about $$ n^{P} + x \pmod P? $$ What can you say about $$ n^{2P-1} + x \pmod P? $$ $$ n^{3P-2} + x \pmod P? $$ https://en.wikipedia.org/wiki/Fermat%27s_little_theorem
Example: $n=2,x=3,P=5.$ $$ 2 + 3 = 5 \equiv 0 \pmod 5 $$ $$ 2^5 + 3 = 32+3 = 35 \equiv 0 \pmod 5 $$ $$ 2^9 + 3 = 512+3 = 515 \equiv 0 \pmod 5 $$ $$ 2^{13} + 3 = 8192+3 = 8195 \equiv 0 \pmod 5 $$