Problems & Puzzles: Puzzles

Puzzle 838. Follow up to Puzzle 837. In puzzle 837 we asked for prime Kaprekar numbers. Jan van Delden and Emmanuel Vantieghem demonstrated that all prime repunits 1 mod 9 are Kaprekar numbers, but it could happen that exist some prime Kaprekar numbers out of these prime repunits. "...there might exist prime divisors  d  of  N =10^n - 1  that are not repunits but that satisfy  d = 1 (mod N/d). These numbers would be Kaprekar numbers when they are unitary divisors". Q.Find one (if the smallest the better) of these prime Kaprekar numbers out of the prime repunits, or show that these do not exist for some other reason.

---