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

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Rainer Rosenthal

Posts: 963
Registered: 12/6/04
Re: Primes...
Posted: Jan 30, 2003 6:19 PM
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Jim Ferry wrote

> A difficult open problem is whether the limit of
> sqrt(P_(n+1)) - sqrt(P_n) is 0.


Just submitted to the OEIS:

%I A000001
%S A000001 4,2,3,2,4,2,5,3,2,6,2,4,7,4,3,3,8,3,5,9,3,5,4,3,5,11

%N A000001 Smallest number whose reciprocal fits in the square-root gap of
consecutive primes
%C A000001 A difficult open problem is whether the limit of
sqrt(P_(n+1)) - sqrt(P_n) is 0.
(Jim Ferry in sci.math 30-th of January 2003)
%D A000001 Message-ID: <HZc_9.16064$Vf3.180654@vixen.cso.uiuc.edu> Jim Ferry
in sci.math
%F A000001 a(n) = ceiling(1/(w'-w)) where w=sqrt(p(n)) and w'=sqrt(p(n+1))
%e A000001 a(3) = 3 because p(3)=5, p(4)=7, w=sqrt(5) w'=sqrt(7) and
1/(w'-w)=2.44.
%Y A000001 A000040
%O A000001 1
%K A000001 ,easy,nice,nonn,
%A A000001 Rainer Rosenthal (r.rosenthal@web.de), Jan 30 2003

The next interesting sequence is:

1, 113, 1327, ... ??? where a(n) = smallest natural such that
sqrt(P_(k+1)) - sqrt(P_k) < 1/n for all k with P_k > a(n)

I don't have a CAS and would be glad to see someone submit this sequence
with more and better consolidated entries.

Rainer Rosenthal
r.rosenthal@web.de




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