I'm doing some exam practice questions and I am totally stuck on this one, been racking my brain for days without much progress. I would truly appreciate some help.
I tried so many different routes. But my tutor gave this following hint, but I am struggling to proceed with it, or really how it would even lead to a solution:
Start with the simplest example such that $10^n+3$ is composite with $n$ divisible by a small prime factor. I see that:
$10^4\equiv -3 (mod \space 7)$ (2 being the small prime factor obviously) or
$10^9\equiv -3 (mod \space 23)$ (with 3 as a small prime factor)
Then he said, prove that for all $n$ > some number, that it is divisible by this small prime factor.
So using the case for $n=9$, letting $n = 3k$ for some integer $k$, I re-write this as:
$(10^{3})^k\equiv -3 (mod \space 23)$
I know $10^3\equiv -3 (mod 17)$, but I am not sure how to proceed?
I'm usually pretty good with at least going in a reasonable direction with a hint, but I am truly stuck here.
Sincere thanks in advance.