While playing with programs to determine primes and relative
primes, I stumbled over an interesting (at least to me) fact. While the
probability of a random number being prime decreases as the range of possible
random numbers increases (Prime Number Theorem), the probability of two
random numbers being relatively prime is 60.8% Is this something that
is either well known by or trivially obvious to prime number gurus?
That there should be such a constant is "obvious", finding its value takes
more work. In 1849 Dirichlet showed that the probability is 6/2
roughly as follows.
Suppose you pick two random numbers less than n, then
[n/2]2 pairs are both divisible by 2.
[n/3]2 pairs are both divisible by 3.
[n/5]2 pairs are both divisible by 5.
(Here [x] is the greatest integer less than or equal to x,
usually called the floor function.) So the number of relatively prime
pairs less than or equal to n is (by the inclusion/exclusion principle):