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Consider prime constellations $p,p+2s$ where both $p,p+2s$ are prime.

For instance for $s=1$ we get the twin primes.

We define the counting function $\pi_{2s}(n)$ to count the number of such pairs $p,p+2s$ below or equal to $n$.

Does this conjecture hold ?

$$ \pi_{2a}(n+(6a+4)^3)+(6a+4)^3 > \pi_{4a}(n)$$

for all positive integer $n>2$ and all positive integer $a>0$ ?

I know that according to Hardy-Littlewood this probably is false, but has any counterexample actually been found ?

I know the skewes numbers for (cousin primes) $\pi_4 >$ the skewes number for (prime twins) $\pi_2$ and I know the skewes number for sexy primes is unknown if it even exists.

($1369391$ for twins vs $5206837$ for cousins )

see Wikipedia entry on Skewes numbers (for prime k-tuples).

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It is false, try $a=1, n=250\,003\,639$.

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  • $\begingroup$ Wow that is a large number, larger than both related skewes numbers and the prime cousin one was only found in 2019 by Toth. So the prime cousins catch up the prime twins and both are above the Hardy-Littlewood estimate , right ? Or did they go under the Hardy-L estimate again ? How did you find this number ? Theory ? Computer code ? $\endgroup$
    – mick
    Commented Jun 28, 2023 at 22:07
  • $\begingroup$ @mick It needed to be at least somewhat large, since you were asking for cousin primes to not just outpace twin primes but to do so by 1000 (and to give the twin primes a range longer by 1000, but that doesn't make such a difference). My approach was purely computational and does not find the smallest candidate, though I imagine it's not far off. $\endgroup$
    – Charles
    Commented Jun 29, 2023 at 2:00
  • $\begingroup$ Right. Thanks. But I can not confirm it myself. Maybe if others confirm or you add the codes etc. Im willing to believe you but ... $\endgroup$
    – mick
    Commented Jun 29, 2023 at 22:40
  • $\begingroup$ I was able to find a smaller example by looking more carefully (prime-by-prime instead of in blocks of 1 million). $\endgroup$
    – Charles
    Commented Jun 30, 2023 at 12:32

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