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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.

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up vote 6 down vote accepted

Yes. Take q sufficiently big and fixed (in terms of a and b). Then the question is, is there some prime p between qa and qb? Use the prime number theorem to estimate pi(qb) - pi(qa) > 0, where q is chosen to be big enough so that the main term is bigger than the error terms. QED.

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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.)

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It's enough to assume (by contradiction) that for any prime q, there are no primes between qa and qb. Then the conclusion follows without any complication if a=1, but I can't still figure out how you manage the case a=!1. –  Gian Maria Dall'Ara Oct 23 '09 at 18:29
    
Hmm. I might have to take this one back; this works fine if q is allowed to be arbitrary but for q prime the intervals [qa, qb] might skip all the intervals where primes are close together. –  Qiaochu Yuan Oct 23 '09 at 18:52
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