TITLE: Density of a subset of the reals
QUESTION [7 upvotes]: The rationals are clearly dense in the real number system, i.e. for every pair a < b of real numbers there exists a rational number p/q s.t. a < p/q < b. I conjecture the same to be true with p and q both primes. Any idea of how one could prove it? It should depend on some strong result on the distribution of prime numbers.

REPLY [3 votes]: You don't need the prime number theorem.  Suppose the result is false, i.e. for fixed a and b there are only finitely many q such that there is a prime between qa and qb.  Then the nth prime pn grows at least as fast as (b/a)^n; in particular, sum 1/pn converges, which we know to be false (and which is totally elementary).  (This argument is slightly thorny to make rigorous, but it's just a matter of handling constants.)