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-----Message d'origine-----

De : Didier van der Straten [mailto:vdstrat@...]

Envoyé : jeudi 25 décembre 2003 22:17

À : Paul Leyland

Objet : RE: [PrimeNumbers] N-th prime boundary?

Allow me to doubt about this response N/(ln N - 2). This seems to approach

the "prime count function" up to N, while I believe the question is how to

approach

the Nth prime, with a value slightly larger than the actual Nth prime,

which would suggest a formula such as N * (ln N + k). Finding a "good" k

might be

another challenge, including checking it for high values of N.

Or I obviously missed one point. Forgive me... at Christmas or at New Year's

eve.

Didier

-----Message d'origine-----

De : Paul Leyland [mailto:pleyland@...]

Envoyé : mardi 23 décembre 2003 18:05

À : William Bouris; primenumbers@yahoogroups.com

Objet : RE: [PrimeNumbers] N-th prime boundary?

I believe that N/(ln N - 2) will serve your purpose.

Paul

> -----Original Message-----

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> From: William Bouris [mailto:melangebillyb@...]

> Sent: 23 December 2003 16:16

> To: primenumbers@yahoogroups.com

> Subject: [PrimeNumbers] N-th prime boundary?

>

> Hello, Group.

>

> Is there a simple formula for calculating an upper boundary

> which would always contain a particular nth-prime number?

> ie. input n --> formula ---> calculated limit > prime p. I

> bet that this question has been asked several times, but I

> just need a "slightly larger than p" value for any given n.

>

> Thanks in advance, Bill

>

>

>

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http://docs.yahoo.com/info/terms/ - I found the following sci.math posting which may be of use:

--------------------------------------------------------------

From: bobs@...

Subject: Re: Upper bounds for the nth prime number

Date: 1999/03/18

The Rosser & Schoenfeld results, as improved by Robin are the best known.

p_n > n log n + n (log log n - 1.0072629)

for n > 7021, p_n <= nlog n + n (log log n - .9385)

-------------------------------------------------------------

I don't know a reference for this. Perhaps it can be found in the

following paper I found on MathSciNet.

Massias, Jean-Pierre(F-LIMOS-OC); Robin, Guy(F-LIMOS-OC)

Bornes effectives pour certaines fonctions concernant les nombres

premiers. (French. French summary) [Effective bounds for certain functions

involving primes] J. Thor. Nombres Bordeaux 8 (1996), no. 1, 215--242.

Abstract:

Let $p_k$ denote the $k$th prime number and let

$\theta(p_k)=\sum^k_{i=1}\log p_i$ be the Chebyshev function. The authors

obtain new explicit upper and lower bounds for the expressions $p_k$,

$\theta(p_k)$, $\sum^k_{i=1}p_i$ and $\sum_{p\leq x}p$, thereby improving

several estimates obtained by J. B. Rosser and L. Schoenfeld\

\ref[Illinois J. Math. 6 (1962), 64--94; MR 25 #1139; Math. Comp. 29

(1975), 243--269; MR 56 #15581a] and Robin \ref[Acta Arith. 42 (1983),

no. 4, 367--389; MR 85j:11109].

----------------------------------------------------------------

---Edwin

------------------------------------------------------------

W. Edwin Clark, Math Dept, University of South Florida,

http://www.math.usf.edu/~eclark/

------------------------------------------------------------ - At 05:46 AM 12/26/2003, Didier van der Straten wrote:

>Allow me to doubt about this response N/(ln N - 2). This seems to approach

I'm not quite sure what you're asking. Are you asking for an upper bound

>the "prime count function" up to N, while I believe the question is how to

>approach

>the Nth prime, with a value slightly larger than the actual Nth prime,

>which would suggest a formula such as N * (ln N + k). Finding a "good" k

>might be

>another challenge, including checking it for high values of N.

on the nth prime number? A number that is slightly larger than the nth

prime? Something else?