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

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Jim Ferry

Posts: 542
Registered: 12/6/04
Re: Primes...
Posted: Jan 30, 2003 12:09 PM
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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 |




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