# Combining primes for getting primes?

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?

• $711$ is not a very good example since it equals $79\cdot 9$ – 5xum Jun 22 '14 at 9:52
• 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?). – barak manos Jun 22 '14 at 9:53
• @5xum it was nt for prime it was just to introduce the notation – Shivam Patel Jun 22 '14 at 9:54
• There is a small typo, it should be $q=11$. – Indrayudh Roy Jun 22 '14 at 9:55
• 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 $m$. – Yves Daoust Jun 22 '14 at 10:26

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?