Problems & Puzzles: Puzzles

 Puzzle 795. Primes reversing substrings Emmanuel Vantieghem sent the following nice puzzle For  a = 0, 1, 2, ... , 8, 9, consider the following transformations F(a, m)  of a positive integer  m : Look where the digit  a  occurs in  m.  Then get  F(a, m)  by reversing the digits between two successive appearances of  a. Example : m = 12346597804454672013254628. If we take  a = 2, we should reverse the strings 34659780445467, 013 and 546, so : F(2,m) = 12764544087956432310264528 The digit  9  occurs only once in  m, hence ; F(9,m) = m.   My question is :find primes  p  with the property that, for each used digit a  in  p,  F(a, p) is a prime different from p ?   The smallest prime using three digits is  p = 1051051. Indeed, p uses the digits  0, 1  and  5  and:    F(0,p) = 1015051, prime    F(1,p) = 1501501, prime    F(5,p) = 1050151, prime The minimal such primes for 4, 5 & 6 digits are:  39703709, 1497294217, 102830938219, respectively.   Q. What is the smallest such prime using 7, 8, 9 & 10, ... digits ?

Contributions came from Emmanuel Vantieghem

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Emmanuel wrote on July  31, 2015

The smallest such prime using 7 digits could be 10452765470621.
Using 8 digits it might be 1036482501452863.

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Emmanuel wrote on August 04, 2015

I found a solution that uses nine digits: p = 102756857398639201

And finally, I found a prime p  that uses ten digits (a pandigital prime) such that  F(a,p)  is prime for  a=0, 1, 2, ... , 8, 9: p = 102344856798765493201.
It is very likely the smallest (but I'm not  100% sure).

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