Correct and complete solutions were received from Jeremy Light, Adam Brown.  Correct and complete solutions were also received from Ahron Teitelman, Shekhar Joglekar, Burkart Venzke, Al Zimmermann, Bulat Khaidarov, Duc Duc, Paul Botham, Juan Carlos Marivela, Lou Cairoli.

The one digit dips (allowing the digit 1) are the primes 1, 2, 3, 5, 7.  A two digit dip is a two digit prime whose digits are themselves prime, namely, 11, 13, 17, 23, 31, 37, 53, 71, 73.  Any three digit dip must be a three digit prime whose first two digits and last two digits are dips, giving only 131, 137, 173, 311, 313, 317, 373.   Similarly, any four digit dip must be a four digit prime whose first three digits and last three digits are three digit dips, leaving only 3137 and 1373.  Any five digit dip would have these four digit dips as its first four and last four digits, making 31373 the only possibility.  (Un)fortunately, 31373 = 137 × 229 is not prime!  So there are no five digit dips, and consequently no dips with more than five digits.   

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ã2003 Alberto L. Delgado