It turns out that the density of primes is such that
the number of primes below some number X
is proportional to X divided by the natural log of X.
So this is log base E.
And I won't attempt to prove this.
I'll resort to proof by intimidation.
This was conjectured by Gauss and then Legendre
and proven later.
And that means that the probability--if we pick some random number,
the probability that that number is prime
is approximately 1 over the natural log of X.
So the question is: how many guesses do we expect to need
to find a prime number that's around 100 decimal digits long?
And in computing this, you should assume that this probability
that a random X is prime is equal to 1 over the natural log of X
even though this is an approximation.
