20170208, 01:28  #1 
"Rashid Naimi"
Oct 2015
Remote to Here/There
3·643 Posts 
Bonse's inequality
Hi all,
https://en.m.wikipedia.org/wiki/Bonse%27s_inequality It seems to me that the inequality can be true for higher powers (if not any given higher power), for an appropriately higher (lower) bound for "n". Any thoughts, proofs, counter proofs our insights are appreciated. Thank you in advance. Last fiddled with by a1call on 20170208 at 01:32 
20170208, 01:43  #2 
"Rashid Naimi"
Oct 2015
Remote to Here/There
789_{16} Posts 
In particular I am interested in estimates (or preferably lower bound) for n for:
(p1)# < p^n 
20170208, 01:48  #3  
"Forget I exist"
Jul 2009
Dumbassville
8,369 Posts 
Quote:


20170208, 06:14  #4 
Aug 2006
13454_{8} Posts 
(p1)# and p# are around e^p, so n would need to be around p/log p for that to work. In particular, for any fixed n, (p1)# is larger than p^n for large enough p.
Last fiddled with by CRGreathouse on 20170208 at 06:15 
20170208, 09:25  #5 
"Rashid Naimi"
Oct 2015
Remote to Here/There
3×643 Posts 
Thank you very much for the replies. In particular Mr Greathouse for answering my particular question.

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