MHB Does the Inequality Involving Sums of Consecutive Twin Prime Pairs Always Hold?

AI Thread Summary
The discussion centers on the inequality involving sums of consecutive twin prime pairs, specifically whether p_n + p_{n + 1} is always greater than or equal to p_{n + 2} + p_{n - 2} for all n ≥ 4. While asymptotic analysis suggests that the inequality holds in general, counterexamples, particularly for n = 29, indicate that it may not always be true. The participants explore the possibility of a threshold n_0 beyond which the inequality might hold, but skepticism remains regarding its validity. Overall, the conversation highlights the complexity of prime number behavior and the challenges in establishing definitive inequalities. The inquiry into the inequality remains open and unresolved.
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Let \ \ p_n \ \ = \ \ the \ \ nth \ \ prime \ \ number.Examples:p_1 \ = \ 2

p_2 \ = \ 3

p_3 \ = \ 5

p_4 \ = \ 7- - - - - - - - - - - - - - - - - - - - - - - - - - - - Let \ \ n \ \ belong \ \ to \ \ the \ \ set \ \ of \ \ positive \ \ integers.
Prove (or disprove) the following:p_n \ + \ p_{n + 1} \ \ \ge \ \ p_{n + 2} \ + \ p_{n - 2}, \ \ \ for \ \ all \ \ n \ \ge \ 4.Examples:\ \ 7 \ + \ 11 \ \ > \ \ 13 \ + \ \ 3

19 \ + \ 23 \ \ = \ \ 29 \ + \ 13
 
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checkittwice said:
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Let \ \ p_n \ \ = \ \ the \ \ nth \ \ prime \ \ number.Examples:p_1 \ = \ 2

p_2 \ = \ 3

p_3 \ = \ 5

p_4 \ = \ 7- - - - - - - - - - - - - - - - - - - - - - - - - - - - Let \ \ n \ \ belong \ \ to \ \ the \ \ set \ \ of \ \ positive \ \ integers.
Prove (or disprove) the following:p_n \ + \ p_{n + 1} \ \ \ge \ \ p_{n + 2} \ + \ p_{n - 2}, \ \ \ for \ \ all \ \ n \ \ge \ 4.[/tex]Examples:\ \ 7 \ + \ 11 \ \ > \ \ 13 \ + \ \ 3

19 \ + \ 23 \ \ = \ \ 29 \ + \ 13

Asymtotically \(p_n+p_{n+1} \sim (2n+1)\log(x)\) and \(p_{n-2}+p_{n+2} \sim 2n\log(n)\).

so the inequality eventually holds, and how many terms we need to check explicitly before we can rely on the asymtotics can probably be determined from (on second thoughts it can't, the inequalities have too wide a spread):

\(n\log(n)+n\log(\log(n))-n<p_n<n\log(n)+n\log(\log(n)), \ \ n\ge 6\)

It appears to fail for \(n=29\), when \(p_n=109,\ p_{n+1}=113,\ p_{n-2}=103,\ p_{n+2}=127\)

There are also plenty of other counter examples for primes less than \(10^6\).

So, the new question is: Is there a \(n_0\) such that for all \(n>n_0\) the inequality holds? I would guess the answer is no.

CB
 
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