# Golbach's partitions: is there always one common prime in $G(n)$ and $G(n+6)$ , $n \ge 8$ (or a counterexample)?

I am trying to find a counterexample for the following expression when $d=6$. ($G(n)$ = Goldbach partition of the even number $n$)

${\forall}$ n=2*k / k${\in}$N, n${\geq}$8

${\exists}$(${p_i}$,${p_j}$) /
(${p_i}$,${p_j}$) ${\in}$ G(n),
(${p_i}$,${p_j}$+d) ${\in}$ G(n+d)

for a fixed distance $d=2*q\ /\ q$${\in}\Bbb N, d\geq 2 It means that the Goldbach partitions of the even number n, and the even number n+d, at least contain one common prime {p_i}, being the counterpart primes {p_j} and {p_j}+d, and d is a fixed even distance, for instance always 2, always 4, always 6, etc. so: {p_i}+{p_j}$$=n$
and
${p_i}$+(${p_j}$+d)$=n+d$

I have found very quickly counterexamples when the distance $d$ is set to $2$,$4$ or $8$:

$d=2 n=38 n+2=40$ do not have any common primes in $G(38),G(40)$

$d=4 n=172 n+4=176$ do not have any common primes in $G(172),G(176)$

$d=8 n=38 n+8=46$ do not have any common primes in $G(38),G(46)$

But in the case $d=6$, I have run a test for the first $9000$ even numbers (after that point my computer slows down very much) and I did not find a counterexample (assuming the code is correct), so in the case of n and n+6, I have found always at least one common prime in $G(n)$ and $G(n+6)$, and the counterparts are a couple of sexy primes at least for any even number $n$ between $[8,9000]$. So another way of saying this is that there is always a couple of sexy prime shared between $G(n)$ and $G(n+6)$ when the even number $n$ is greater or equal to $8$.

I am very curious because for $d=2,4,8$ was very easy to find a quick counterexample, but not in the case of d=6 probably because is greater than the $9000^{th}$ even number or a test error (I am assuming I did correctly my test, but it could happen).

UPDATE 2015/04/09

Note: all the explanation above was assuming that:

$\ \ n=p+q\ /\ p\le q$ and

$\ \ n+6 = p+(q+6)$,

But there is also the possibility that the common prime $p\gt q$ (*).
In that case, the sexy primes are $q$ in the list of primes of $G(n)$
and $q-6$ in the list of primes of $G(n+6)$.

Below there is an extension of the explanation (for the cases in which $p\le q$, but same idea would apply for (*) $p\gt q$):

For any $n≥8$ even seems to exists at least a prime pair $(p,q)\ /\ n=p+q$, in which $p$ or $q$ has a sexy prime counterpart, e.g $q+6$, also prime, and $n+6=p+(q+6)$.

E.g. $n=8=3+5$ and $n+6=14=3+11$, where the ‘shared’ or ‘fixed’ prime
in $n=8$ and $n=14$ is $3$ and the sexy primes are $5$ in $n=8$ and
$11(=5+6)$ in $n=14$.

E.g.: $n=10=3+7$ and $n+6=16=3+13$, so the ‘shared’ prime is $3$ in
$n=10$ and $n=16$ and the sexy primes are $7$ in $n=10$ and $13$ in
$n=16$.

E.g.: $n=17890=17837+53$ and $n+6=17896=17837+59$, in the same way
than above.

I still did not find counterexamples, every Goldbach partition of a given even number $n$ and $n+6$ seems to cover this property.

E.g. this is a picture of a visualization of the list of Goldbach primes for each $n \in [466..480]$:

Each column is the list of Goldbach primes for the even number $n$ written at the top, and the primes below (top-down) are in ascending order the primes $p$ able to sum up $n=p+q$, where $q$ is also prime.

In the picture there are samples of the common prime behavior, in different colors, for the following even numbers $(n,n+6)$:

$(n=466$,$n=472)$

$(n=468$,$n=474)$

$(n=470$,$n=476)$

$(n=472$,$n=478)$

$(n=474$,$n=480)$

E.g.
$n=466=5+461$ and $n=466+6=472=5+(461+6)=5+467$, so (a) in this example $p=5$ is in the list of Goldbach primes of $n=466$ and $n+6=472$, and (b) the sexy prime $q=461$ belongs to the list of $n=466$, and its counterpart sexy prime $q+6=467$ belongs to the list of Goldbach primes of $n=472$.

But it seems it does not matter which random even $n\ge8$ is taken, always the list of Goldbach primes for the even number $n$ shares a prime $p$ with the list of $n+6$ and there is a couple of sexy primes $(q,q+6)$ (if $p \le q$) or $(q,q-6)$ (if $p \gt q$) shared between $n$ and $n+6$, $q$ belongs to the list of $n$ and $q+6$ (if $p \le q$) or $q-6$ (if $p \gt q$) belongs to the list of $n+6$.

#### Solutions Collecting From Web of "Golbach's partitions: is there always one common prime in $G(n)$ and $G(n+6)$ , $n \ge 8$ (or a counterexample)?"

I believe you are asking if, for each even $n\ge8,$ there is a prime $p$ such that $n-p$ and $n-p+6$ are both prime. I verified that this is true for all $n\le10^8$ by finding appropriate $p$. Heuristically $n$ fails with probability
\begin{align} &\prod_{3\le p\le n-3}1-\frac{1}{\log(n-p)} \cdot \frac{1}{\log(n-p+6)}\\ \approx&\prod_{3\le p\le n-3}1-\frac{1}{\log^2n}\\ =&\ \exp\sum_{3\le p\le n-3}\log\left(1-\frac{1}{\log^2n}\right)\\ \approx&\ \exp-\!\!\!\!\!\!\sum_{3\le p\le n-3}\frac{1}{\log^2n}\\ =&\ \exp\frac{1-\pi(n)}{\log^2n}\\ \approx&\ \exp\frac{-n}{\log^3n} \end{align}
and so the ‘probability’ that it fails for $n>10^8$ is bounded above by
$$\int_{10^8}^{+\infty} \exp\frac{-x}{\log^3x} \text d x<10^{-6944}.$$