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](https://en.wikipedia.org/wiki/Skewes%27s_number) (for prime k-tuples).



Prime gap conjecture $ \pi_{2a}(n+(6a+4)^3)+(6a+4)^3 > \pi_{4a}(n)$ counterexamples?
It is false, try $a=1, n=250\,003\,639$.


