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[Number Theory] pi(X,x,x+2)+pi(X,x,x+4)
Let pi(X,x,x+2) be the amount of prime twins between 2 and X.
Likewise let pi(X,x,x+4) be the amount of prime constellation(x,x+4) ( or primes with gap 4 also called "cousin primes") between 2 and X.

Since the constellation (x,x+2,x+4) does not exist for x>3 :
( one of x,x+2,x+4 must be a multiple of 3 )

I had the "fantasy" that the two constellations (x,x+2) and (x,x+4) "repel" each other.

( maybe Im doing to much physics lately :p )

1) the Riemann Hypothesis
2) The largest prime gap is =< O(ln(X)^3) (thus a weaker form of cramer's conjecture)
3) The first Hardy-Littlewood conjecture for prime constellations
( the one that says prime twins and prime cousins are well estimated by C_2 Li_2(x) )

my " fantasy " lead me to the intuition that more often then not , for large X ,

0.5 (pi(X,x,x+2)+pi(X,x,x+4)) is closer to C_2 Li_2(x) then either of pi(X,x,x+2) or pi(X,x,x+4).

This is the main idea.

Variations exist such as other types of averages between the countings of twins and cousin primes.
And these other averages lead me to tetration.

But lets start with the main idea.
Has anyone ever seen this or a similar conjecture ?
What are your thoughts on this ?
Does my intuition make sense ?
Does it agree with " numerical evidence " ?
Are there " strong statistical arguments " for or against it ?
I assumed 3 things as explained above. Are those the best assumptions ? Are more needed ? Less or other ?
What do you think ?

BTW for those intrested in another conjecture for prime constellations see :


The idea that complicated functions can be better described by simpler ones simply by taking an average (or other simple connections) gives me the feeling of " mathematical beauty ". Comparable to the PNT.

I have been trying to prove the conjecture of the OP by looking deep into proofs of PNT but without succes sofar.



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