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I was thinking about twin primes and I came to ask this question:

If we have two distinct primes $P_1$ and $P_2$ which are both greater than $3$, then does there always exist a prime $P_3$ such that

$$\begin{cases}P_1+6 P_3 \in \mathbb P \\ P_2+6 P_3 \in \mathbb P \\ \end{cases}$$

I tested this for all prime pairs up to $3581$ (500th prime) and the only one for which the $P_3$ was greater than $541$ (100th prime) was pair $\{1451,3163\}$ (230th and 447th primes). But $P_3 = 631$ works.

I might add that this is, to my mind, equivalent to twin prime conjecture. Why? Well every twin prime pair is of the form $(6k-1,6k+1).$ Now, for some given prime $P$, this can be written as $$(6(n P+v)-1,6(n P+v)+1)$$ and this, by expansion, is $$(6 n P+6 v-1,6 n P+6 v +1).$$ But there is a potential twin prime here too, namely $(6v-1,6v+1).$ So by setting $n=1,$ and $$\begin{cases}6 v-1 \in \mathbb P \\ 6v+1 \in \mathbb P \\ \end{cases}$$ which is clearly possible, then, if the above is true, we can select $P$ so that both of the originals are prime and we have gotten ourselves a new and bigger twin prime pair. So then there would be infinitely many twin primes.

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Is it even known whether for every prime $P_1>3$ there exists a prime $P_3$ such that $P_1+6P_3 \in \Bbb P$? –  Jaycob Coleman Oct 24 '13 at 20:33
    
If you require this only for $P_1=5$ and $P_2=7$ while at the same time generously allowing natural numbers for $P_3$, you have the twin prime conjecture. –  Hagen von Eitzen Oct 24 '13 at 20:56
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@HagenvonEitzen I don't see how this is the case when he only conjectures the existence of at least one such $P_3$ for each pair. –  Jaycob Coleman Oct 24 '13 at 21:09
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