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$\begingroup$ moonface: I'm a bit confused, here. Certainly heuristics given by the PNT would show that for most primes you have p^{1+\epsilon}, but I don't think this rules out the possibility that there are infinitely many primes for which it's not true. (Certainly it doesn't trivially rule out the possibility!) $\endgroup$

Harrison Brown
– Harrison Brown

2009-10-18 11:47:08 +00:00
Commented Oct 18, 2009 at 11:47
2

$\begingroup$ (This might show up twice by accident, sorry.) Let e > 0. Mindlessly applying the heuristic, the chance f(p) that "there is no prime less than p^{1+e} in the progression" decreases very fast with p; in fact, the sum of f(p) over all p converges, suggesting this event happens only finitely often. For an example of the limitations of such reasoning, see "Primes in short intervals" by H. Maier. $\endgroup$

moonface
– moonface

2009-10-18 17:51:50 +00:00
Commented Oct 18, 2009 at 17:51

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