Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
[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.



Possibly Related Threads...
Thread Author Replies Views Last Post
  Hyper operators in computability theory JmsNxn 5 7,436 02/15/2017, 10:07 PM
Last Post: MphLee
  Cellular auto : rule 30 number ? tommy1729 0 2,099 08/03/2016, 08:31 PM
Last Post: tommy1729
  Set theory debate : cantor 1st / Virgil argument. tommy1729 1 3,139 12/08/2015, 11:14 PM
Last Post: tommy1729
  [2015] Spiderweb theory tommy1729 0 2,724 03/29/2015, 06:25 PM
Last Post: tommy1729
  [number theory] sieving with a_i mod p_i tommy1729 7 11,295 09/12/2014, 07:28 AM
Last Post: tommy1729
  " fake ring theory " tommy1729 0 2,871 06/11/2014, 11:29 PM
Last Post: tommy1729
  the " average restart " of a number. tommy1729 0 2,309 03/26/2014, 01:02 AM
Last Post: tommy1729
  A conjecture about number theory and tetration tommy1729 0 2,782 10/23/2013, 09:28 PM
Last Post: tommy1729
  (number theory) How tommy1729 does it. tommy1729 0 2,802 08/12/2013, 09:13 PM
Last Post: tommy1729
  Number theory and hyper operators JmsNxn 7 11,119 05/29/2013, 09:24 PM
Last Post: MphLee

Users browsing this thread: 1 Guest(s)