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Old 2017-02-08, 01:28   #1
a1call
 
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"Rashid Naimi"
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Default 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 2017-02-08 at 01:32
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Old 2017-02-08, 01:43   #2
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In particular I am interested in estimates (or preferably lower bound) for n for:

(p-1)# < p^n
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Old 2017-02-08, 01:48   #3
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Quote:
Originally Posted by a1call View Post
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.
http://math.stackexchange.com/questi...first-n-primes suggest a proof could be as simple as noting things like there's always a prime between n and 2*n and then the fact that 2*...*p(n) has 1 product we can take away 2*p(n) we know this is larger than the base if we can then prove that 1/2*pn*1/4*pn*.....3 > p(n+1) we can prove that the statement is always true. that p(n+1)^2 < 2*...*p(n)
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Old 2017-02-08, 06:14   #4
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Quote:
Originally Posted by a1call View Post
In particular I am interested in estimates (or preferably lower bound) for n for:

(p-1)# < p^n
(p-1)# 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, (p-1)# is larger than p^n for large enough p.

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Old 2017-02-08, 09:25   #5
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Thank you very much for the replies. In particular Mr Greathouse for answering my particular question.
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