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Let $n$ be a natural number. Let $\pi_k$ be denoted as follows.

$ \pi_{2}(n) = $ the number of twin primes $(p, p+2)$ with $p \le n$.

$ \pi_{4}(n) = $ the number of cousins primes $(p, p+4)$ with $p \le n$.

$ \pi_{6}(n) = $ the number of sexy primes $(p, p+6)$ with$ p \le n$.

According to Mathworld “Prime Constellation”, asymptotic expressions (conjectures) of $\pi_{2}(n)$ and $\pi_{4}(n)$ are the same. They equal $1.32032..\int_{2}^{n} dx/(log(x))^2$. That of $\pi_{6}(n)$ is twice the integral. Is there any simple explanation or reason of it?

SeiichiKirikami
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    The article you cite doesn't say anything like the statement you attribute to it. It clearly specifies that ``The first Hardy-Littlewood conjecture states that the numbers of constellations...'' (emph added). Please refrain from inaccurately ascribing statements to conscientiously written exposition, and make sure you exercise due diligence before posting. – stochasticboy321 Jun 24 '17 at 09:13
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    @stochasticboy321 That's what it says. See (2) and (4) under First Hardy-Littlewood Conjecture. You're attitude and criticism is way off base. – user56983 Jun 24 '17 at 18:41
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    @PokerFace It seems like you didn't quite read what I've written. My point is that the page does not claim that the equations are true, it only claims that they are conjectured to be true. That's a world of difference in mathematics. Now, especially for a problem this famously open, to casually say that some bit of text claims its solution is egregious, & implies that the poster did not do the work expected of them before posting. What makes this particularly irritating is that the Mathworld articles are carefully and clearly written, & inaccurately putting words in their mouth is just wrong. – stochasticboy321 Jun 24 '17 at 19:01
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    It seems like you just want to vent about something completely irrelevant. – user56983 Jun 24 '17 at 19:11
  • @stochasticboy321 I know that they are conjectures based on experiments. I found an interesting literature and posted it in Math Stack Exchange. https://math.stackexchange.com/questions/2300762/twin-primes-by-an-amateur-mathematician I hope that Yasui’s 1st inequality for twin primes will be proved or disproved for the infinity. Then the infinity of cousins primes will be proved or disproved in a similar manner. – SeiichiKirikami Jun 24 '17 at 23:10
  • The last line of my comment above is meant as follows. The cousins primes have also an inequality corresponding to Yasui's 1st inequality. This inequality will be proved or disproved. – SeiichiKirikami Jun 25 '17 at 00:21

3 Answers3

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The Hardy–Littlewood conjecture says in particular that the twin prime counting function $\pi_2(x)$ has the asymptotics $$ \pi_2(x)\sim 2C_2 \frac{x}{\log (x) ^2}, $$ for $C_2=2\prod_{p\ge 3}\frac{p(p-2)}{(p-1)^2}\sim 0.66016...$, and $$ \pi_4(x)\sim 2C_4 \frac{x}{\log (x) ^4}, $$ with $C_4=\frac{27}{2}\prod_{p\ge 5}\frac{p^3(p-4)}{(p-1)^4}$.

Dietrich Burde
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  • For much more detail and a substantially better approximation, see Sebah and Gourdon (2002): http://numbers.computation.free.fr/Constants/Primes/twin.pdf. – DanaJ Jun 26 '17 at 18:50
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No such asymptotic expression is known. If you had known one, you would have solved the Twin Prime conjecture which states that There are infinitely many twin primes.

Dietrich Burde
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Meet Taraviya
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If we set up the sieve $\Delta \mathbb{P}_6$, observe that there are two ways to form the odd primes with a gap of 6.

$$(1)\ \ k \in \{ n: \{\{6n-1,6n+5\} \subset \mathbb{P} \} \setminus \{\{6n+1\} \subset \mathbb{P} \} \} $$ and $$(2)\ \ k \in \{ n: \{\{6n+1,6n+7\} \subset \mathbb{P} \} \setminus \{\{6n+5\} \subset \mathbb{P} \} \} $$

We subtract those values for which $\{6n+1\} \subset \mathbb{P}$ in $(1)$ and $\{6n+5\} \subset \mathbb{P}$ in $(2)$, because those coordinates in the k-tuple must be composite so that the gap is between consecutive primes; $\{0,2,6\}$ is a different tuple than $\{0,4,6\}$.

If we consider the heuristic argument, then there is an interesting observation. Consider $\Delta \mathbb{P}_2$, the set of numbers for which $\{6n-1, 6n+1\} \subset \mathbb{P}$ and $\Delta \mathbb{P}_4$, the set of numbers for which $\{6n+1, 6n+5\} \subset \mathbb{P}$.

Goldston has a good introductory level piece that he wrote in 2008, "Are there infinitely many twin primes?" On p.12 of the piece, he explains the justification for the twin primes constant and the expression he refers to as li$_2(x) = \int_2^x{\frac{dt}{\log^2t}}$ which he goes on to describe as the probability that two numbers, being approximately equal are both prime. The constant $C_2$ is described in terms of the dependency between one being prime and the other being prime as well.

Because both expressions $(1)$ and $(2)$ above describe the same relationship independently of each other then we observe doubling between expressions $(2), (4)$ into $(6)$.

user56983
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