I am trying to check if a very big number ($>10^{10,000,000}$) is possibly prime. I have written a computer program to check if the number has any smallish (less than like $600,000,000$) factors... it doesn't. I know that the chance of a random number $p$ being prime is $\dfrac{1}{\ln p}$, but what if it doesn't have any factors less than $600,000,000$? Or more generally, what is the probability that $p$ is prime if it doesn't have any factors less than $x$?
I thought that it might be $\dfrac{\ln(x)}{\ln p}$ but since you only have to check up to the square root of the number to confirm it's prime that didn't make since. I would guess it might be $\dfrac{\ln(x)^2}{\ln p}$ but that's just a guess.
Any help is appreciated. Thanks in advance!