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 June 16th, 2013, 10:19 PM #1 Senior Member   Joined: Apr 2013 Posts: 425 Thanks: 24 True or not Hello! If $p_n,p_{n+1}>1$ where $n\in N*$ are two prime numbers consecutive then there is the inequality $\frac{(n+1)p_{n+1}-np_{n}}{p_{n}^2+p_{n+1}^2}<\frac{n(n+1)}{p_{n}p_{n +1}}$. Thank You!
 June 17th, 2013, 09:34 PM #2 Senior Member   Joined: Apr 2013 Posts: 425 Thanks: 24 Re: True or not Hello! If the inequality $\frac{p_{n+1}}{n}-\frac{p_{n}}{n+1}\leq 2$ is true then the inequality of problem is true. Thank You!
June 18th, 2013, 04:49 AM   #3
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Re: True or not

Quote:
 Originally Posted by Dacu If the inequality $\frac{p_{n+1}}{n}-\frac{p_{n}}{n+1}\leq 2$ is true then the inequality of problem is true.
I haven't checked this reduction but I believe it to be true using a conditional bound $2 \cdot \sqrt{\frac{\log(n)}{n}} + \frac{\log(n)}{n}$ which can be shown to be smaller than 2 initially with any small value and then using the fact that it is decreasing.

June 18th, 2013, 05:10 AM   #4
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Re: True or not

Quote:
Originally Posted by mathbalarka
Quote:
 Originally Posted by Dacu If the inequality $\frac{p_{n+1}}{n}-\frac{p_{n}}{n+1}\leq 2$ is true then the inequality of problem is true.
I haven't checked this reduction but I believe it to be true using a conditional bound $2 \cdot \sqrt{\frac{\log(n)}{n}} + \frac{\log(n)}{n}$ which can be shown to be smaller than 2 initially with any small value and then using the fact that it is decreasing.
Hello!
I don't understand the idea.Please give details.
Thank You!

June 18th, 2013, 05:13 AM   #5
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Re: True or not

Quote:
 Originally Posted by Dacu I don't understand the idea.
What's not to understand? I have shown that this inequality is conditionally true assuming Andrica's conjecture.

 June 18th, 2013, 05:17 AM #6 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: True or not It can be proven with the famous theorem of Hoheisel that there are only finitely many counterexamples.
June 18th, 2013, 05:18 AM   #7
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Re: True or not

Quote:
 Originally Posted by mathbalarka What's not to understand? I have shown that this inequality is conditionally true assuming Andrica's conjecture.
You should definitely state that you're using something like Andrica! That's far beyond our ability to prove at the moment and doesn't even follow from RH.

June 18th, 2013, 05:25 AM   #8
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Re: True or not

Quote:
 Originally Posted by CRGreathouse That's far beyond our ability to prove at the moment and doesn't even follow from RH.
Yes, but I never claimed I had a proof, did I? I showed how this inequality almost surely holds.

Quote:
 Originally Posted by CRGreathouse It can be proven with the famous theorem of Hoheisel that there are only finitely many counterexamples.
Hmm, it never occurred to me. Oh well!

 June 18th, 2013, 06:16 AM #9 Senior Member   Joined: Apr 2013 Posts: 425 Thanks: 24 Re: True or not I do not understand!From $p_{n+1}-p_{n} I do not see how it would result $\frac{p_{n+1}}{n}-\frac{p_{n}}{n+1}\leq 2$. Thank You!
June 18th, 2013, 06:20 AM   #10
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Re: True or not

Quote:
 Originally Posted by Dacu From $p_{n+1}-p_{n} . . .
This is not Hoheisel's result.

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