How can we turn any number into a prime number by simply adding more digits?

Solution 1:

It is known that there is a positive number $\delta$ strictly less than 1 such that, if $n$ is large enough, then there's a prime between $n$ and $n+n^{\delta}$. [I think the state of the art has $\delta=.535$] If $n$ is large enough, then $n$ and $n+n^{\delta}$ start with the same however-many-digits-you-like. So that proves it's always possible, although it doesn't give you a good way to do it.

It is conjectured that there's always a prime between $n$ and $n+C(\log n)^2$ for some positive constant $C$ [$C=2$ may even do, at least for large $n$], but this is way beyong what anyone currently knows how to prove.