the prime number theorem is a theorem from number theory. prime numbers are not distributed evenly across the number range. the theorem formalizes the idea that the probability of hitting a prime number between 1 and a given number becomes smaller, as numbers grow. this probability is about n/ln(n), where ln(n) is the natural logarithm function. this means that the probability of hitting a prime number with 2n digits is about half as likely than with n digits. for example, among the positive integers of at most 1000 digits, about one in 2300 is prime (ln 101000 ≈ 2302.6), whereas among positive integers of at most 2000 digits, about one in 4600 is prime (ln 102000 ≈ 4605.2). in other words, the average gap between consecutive prime numbers among the first "n" integers is roughly ln("n").