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Topic:
Primes...
Replies:
14
Last Post:
Feb 11, 2003 7:41 AM




Re: Primes...
Posted:
Jan 30, 2003 12:09 PM


Jeremy wrote: > > I was idly wondering, if P_n is the nth prime, does: > lim n> infinity P_(n+1) / P_n > exist?
Yes. The limit is 1. Let r_n = P_n / (n log n). The prime number theorem says that the limit of r_n is 1. Therefore the limit of r_(n+1) / r_n is 1. The limit of (n log n) / ((n+1) log(n+1)) is also 1, so the limit of r_(n+1) (n log n) / (r_n (n+1) log(n+1)) is, again, 1. This last expression is just P_(n+1) / P_n.
The limit being 1 is equivalent to the limit of d_n / P_n being 0, where d_n = P_(n+1)  P_n. Assuming the Riemann hypothesis, it has been shown that d_n = O(sqrt(P_n) log(P_n)). Somewhat less tight bounds have been established without RH.
A difficult open problem is whether the limit of sqrt(P_(n+1))  sqrt(P_n) is 0.
 Jim Ferry  Center for Simulation  ++ of Advanced Rockets   http://www.uiuc.edu/ph/www/jferry/ ++  jferry@[delete_this]uiuc.edu  University of Illinois 



