Mathematics

Taniya Crane

1 August, 00:54

The first such distribution found is π (N) ~

N

/

log (N)

, where π (N) is the prime-counting function and log (N) is the natural logarithm of N. This means that for large enough N, the probability that a random integer not greater than N is prime is very close to 1 / log (N). Consequently, a random integer with at most 2n digits (for large enough n) is about half as likely to be prime as a random integer with at most n digits. For example, among the positive integers of at most 1000 digits, about one in 2300 is prime (log (101000) ≈ 2302.6), whereas among positive integers of at most 2000 digits, about one in 4600 is prime (log (102000) ≈ 4605.2). In other words, the average gap between consecutive prime numbers among the first N integers is roughly log (N).[1]