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Assume Goldbach's conjecture. Then for every large enough positive integer $n$ there exists a non negative integer $r$ such that both $n+r$ and $n-r$ are primes. Such an integer $r$ will be called a primality radius of $n$. Let's denote by $r_{0}(n)$ the smallest primality radius of $n$. Considering $r$ ranging from $r_{0}(n)$ to the largest primality radius of $n$, is it true that the quantity $\pi(n+r)+\pi(n-r)$ decreases as $r$ increases? If so, does there exist a universal constant $C$ such that $\pi(n+r_{0}(n))+\pi(n-r_{0}(n)))\lt C.\pi(2n)$? Can we get an estimation of the value of $C$?
Thanks in advance.

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  • $\begingroup$ Do you only want to vary over $r$ which are primality radii? $\endgroup$ Aug 4, 2014 at 17:01
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    $\begingroup$ I may be misunderstanding the question, but if $n=144$, the first three primality radii are $5$, $7$ and $35$ corresponding to $288=139+149=137+151=109+179$, and there are more primes ($157$,$163$,$167$,$173$) between $151$ and $179$ than there are ($113$,$127$,$131$) between $109$ and $137$, so $\pi(n+r)+\pi(n-r)$ increases going from the second to the third primality radius. $\endgroup$ Aug 4, 2014 at 17:30
  • $\begingroup$ @Jeremy Rickard: thank you for the counterexample. Still, it would be rather interesting to know the frequency of such "anomalous" variations of the considered quantity. $\endgroup$ Aug 4, 2014 at 18:15
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    $\begingroup$ As to your second question: $C=2$ works by Bertrand's postulate. Using the prime number theorem, it seems straightforward that any fixed $C>1$ works for $n>n_0(C)$, while $C=1$ probably fails for infinitely many $n$'s. $\endgroup$
    – GH from MO
    Aug 4, 2014 at 18:16
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    $\begingroup$ "...it would be rather interesting to know the frequency of such "anomalous" variations of the considered quantity." So maybe you should do some computing, say, for all $n$ up to 1,000,000, and report back to us on what you find? $\endgroup$ Aug 5, 2014 at 3:40

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