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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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    @FredD$a$nielKline This is also a nice way for other mem$b$ers 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.