## question on (un)boundedness

Expand Messages
• Dear group members: Does anybody know of anything solid to point to indicating whether or not there is a bound on the smallest N(b) for which there is no prime
Message 1 of 3 , Sep 24, 2013
Dear group members:
Does anybody know of anything solid to point to indicating whether or not there is a bound on the smallest N(b) for which there is no prime between b^N(b)-b and b^N(b)?  I imagine I may be able to address this heuristically myself without much bother (and the answer is there is no bound, but of course until I actually try I won't be too sure), but I wonder if there is anything stronger than heuristics on this or it's 'the usual for such number theory questions'.
Jim
• ... It is not an easy question to answer. (2^2-2,2^2) = (2,4) (2^3-2,2^3) = (6,8) (2^4-2,2^4) = (14,16) N(2) = 4 (3^2-3,3^2) = (6,9) (3^3-3,3^3) = (24,27)
Message 2 of 3 , Sep 25, 2013
On 9/25/2013 7:03 AM, primenumbers@yahoogroups.com wrote:
> question on (un)boundedness
> Posted by: "James Merickel"moralforce120@... moralforce120
> Date: Tue Sep 24, 2013 11:17 am ((PDT))
>
> Dear group members:
> Does anybody know of anything solid to point to indicating whether or not there is a bound on the smallest N(b) for which there is no prime between b^N(b)-b and b^N(b)? I imagine I may be able to address this heuristically myself without much bother (and the answer is there is no bound, but of course until I actually try I won't be too sure), but I wonder if there is anything stronger than heuristics on this or it's 'the usual for such number theory questions'.
> Jim

It is not an easy question to answer.

(2^2-2,2^2) = (2,4)
(2^3-2,2^3) = (6,8)
(2^4-2,2^4) = (14,16) N(2) = 4

(3^2-3,3^2) = (6,9)
(3^3-3,3^3) = (24,27) N(3) = 3

(4^2-4,4^2) = (12,16)
(4^3-4,4^3) = (60,64)
(4^4-4,4^4) = (252,256) N(4) = 4
• Empirically it appears there is for any n a finite number of values b (both b and n in {2,3,4,...}) such that the prime preceding b^n is not greater than
Message 3 of 3 , Sep 26, 2013
Empirically it appears there is for any n a finite number of values b (both b and n in {2,3,4,...}) such that the prime preceding b^n is not greater than b^n-b, with for reasonably small n an easy-to-find value that makes a good conjecture for the count.  For n=2 it seems certain there are no such, for example, just empirically.  If this finitude is provable it would settle the question asked.
JGM

From: Kermit Rose <kermit@...>
Sent: Wednesday, September 25, 2013 2:30 PM
Subject: [PrimeNumbers] Re: question on (un)boundedness
On 9/25/2013 7:03 AM, primenumbers@yahoogroups.com wrote:
> question on (un)boundedness
> Posted by: "James Merickel"moralforce120@... moralforce120
> Date: Tue Sep 24, 2013 11:17 am ((PDT))
>
> Dear group members:
> Does anybody know of anything solid to point to indicating whether or not there is a bound on the smallest N(b) for which there is no prime between b^N(b)-b and b^N(b)? I imagine I may be able to address this heuristically myself without much bother (and the answer is there is no bound, but of course until I actually try I won't be too sure), but I wonder if there is anything stronger than heuristics on this or it's 'the usual for such number theory questions'.
> Jim

It is not an easy question to answer.

(2^2-2,2^2) = (2,4)
(2^3-2,2^3) = (6,8)
(2^4-2,2^4) = (14,16) N(2) = 4

(3^2-3,3^2) = (6,9)
(3^3-3,3^3) = (24,27) N(3) = 3

(4^2-4,4^2) = (12,16)
(4^3-4,4^3) = (60,64)
(4^4-4,4^4) = (252,256) N(4) = 4

Your message has been successfully submitted and would be delivered to recipients shortly.