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I was thinking about what would happen when we combine two prime numbers $p$ and $q$ into one number $:pq:$ . Like if $p=5$ and $q=3$ , then $:pq:=53$ . Then if $p=7$ and $q= 11$ then $:pq:=711$ and so on for other $p$ and $q$. It seems so that there is a fairly good chance that the newly obtained number is also a prime. Further examples which we can see are (in no specific order) $$331 ,353 , 223 , 233 ,719 \cdots$$ This motivates me to put forward the following question- Does there exist infinitely many prime numbers $p$ and $q$ such that $:pq:$ is a prime?

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  • $\begingroup$ $711$ is not a very good example since it equals $79\cdot 9$ $\endgroup$ – 5xum Jun 22 '14 at 9:52
  • $\begingroup$ With regards to prime numbers, there is nothing particularly special about the number $10$, which you are using as the representation base of $pq$ (e.g., why not use base $4,6,8,12$, etc?). $\endgroup$ – barak manos Jun 22 '14 at 9:53
  • $\begingroup$ @5xum it was nt for prime it was just to introduce the notation $\endgroup$ – Shivam Patel Jun 22 '14 at 9:54
  • $\begingroup$ There is a small typo, it should be $q=11$. $\endgroup$ – Indrayudh Roy Jun 22 '14 at 9:55
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    $\begingroup$ By Dirichlet's theorem, there are infinitely many primes of the forms $10n+3$ or $10n+7$, or $100n+11$... Anyway, remains to be proven that infinitely many of these $n$ are prime. And by the Green-Tao theorem, there are arbitrarily long sequences of $n$ generating only primes. Anyway, remains to be proven that they contain at least a prime $n$. You can think of Bertrand's postulate, but this is not enough as the arbitrarily long sequence (say of length $m$) may start later than $m$. $\endgroup$ – Yves Daoust Jun 22 '14 at 10:26
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Yes, there are infinitely many such primes. I don't have a theorem with proof but I think pigs would fly before someone could prove me wrong on this.

Go to the OEIS and search for "23, 37, 53, 73, 113, 137, 173, 193, 197". They have a list of ten thousand such primes, though of course that doesn't prove there are infinitely many of them. Much more telling is that the entry's keyword field doesn't have the keyword "fini", which they use to mark sequences they know to be finite.

Denote by $\mathcal{L}$ how many base 10 digits an odd prime $q \neq 5$ has. Then we need to find a prime $p$ such that $10^\mathcal{L}p + q$ is also prime. Given that there are infinitely many primes, it seems highly improbable to me that none of them would satisfy this requirement.

A slightly more interesting question would be: for every odd prime $q \neq 5$ does there exist at least one prime $p$ such that $10^\mathcal{L}p + q$ is also prime?

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  • $\begingroup$ Sourpe can you give a mathematical heuristic for your statement . $\endgroup$ – Shivam Patel Jun 23 '14 at 17:36
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    $\begingroup$ I don't know if this counts as a heuristic to you, but I will try: I believe that for each prime other than 2 or 5, there is at least one other prime which can be prefixed to it so that the concatenation is another prime. If I'm correct in that belief, even if for some primes there is only one other prime that works as such a prefix, since there are infinitely many primes that would still prove there are infinitely many such prime concatenations. $\endgroup$ – Robert Soupe Jun 24 '14 at 1:27
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    $\begingroup$ I have a belief that may be less interesting but easier to prove: I believe there are infinitely many primes to which you can suffix 3 or 7 to make another prime. See for example Sloane's A023238. $\endgroup$ – user153918 Jun 24 '14 at 16:00

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