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Here's a possibly interesting prime puzzle. Call a prime $p$ flirtatious if the sum of its digits is also prime. Are there finitely many flirtatious primes, or infinitely many?

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    I'm interested in seeing whether or not it depends on the base. It's clearly infinite for unary. :P2012-06-28
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    Mersenne primes are flirtatious in binary.2012-06-28
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    Down-voted because answer can easily be found by calculating a few terms and searching OEIS. See Gerry Myerson answer below.2012-06-28
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    @FredDanielKline: Not everyone knows about OEIS (yet). Asking a question like this one is a way to find out about it.2012-06-28
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    @FredDanielKline This is also a nice way for other members of the forum, such as myself, to come across an interesting problem they might otherwise not have heard about.2012-07-09

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These are tabulated at the Online Encyclopedia of Integer Sequences. It appears to be known that there are infinitely many, and a link is given to a recent paper of Harman. Some high-powered math is involved.