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Breakdown an $8$ digit number into successive digits such that each number is a prime and with increasing values to the right. For example, with $23353593$ we have:

  • $2-3-3-5-3593$

  • $2-3-3-53-593$

  • $2-3-3-53593$

  • $2-3-353-593$

  • $23-353-593$

  • $233-53593$

We can see that the number $23353593$ has exactly $6$ breakdowns. My question is, what is the maximum number of breakdowns for an $8$ digit number, and what that $8$ digit number is?

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  • $\begingroup$ Simply write a program to check it out if it is of great importance to you. $\endgroup$ – principal-ideal-domain Aug 21 '15 at 19:53
  • $\begingroup$ @principal-ideal-domain, thanks, but I'm not a computer programmer at all $\endgroup$ – Level- 5c Being Aug 21 '15 at 19:58
  • $\begingroup$ But I don't expect a solution different from brute force. I don't think that mathematical theory will help here a lot. Your definition of a breakdown is so random and also based on the representation of the number and not on the number itself that I don't expect a nice theory behind it. $\endgroup$ – principal-ideal-domain Aug 21 '15 at 20:03
  • $\begingroup$ @principal-ideal-domain, I conjecture that the maximum would not exceed 18 $\endgroup$ – Level- 5c Being Aug 21 '15 at 20:16
  • $\begingroup$ I am quite sure that the first 4 digits of that number must be 2337 $\endgroup$ – Level- 5c Being Aug 21 '15 at 21:34
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The answer is $12$, which is given by the number $23374159$ :

  • $2-3-3-7-41-59$
  • $2-3-37-41-59$
  • $2-3-3-7-4159$
  • $23-37-41-59$
  • $2-3-37-4159$
  • $2-3-3-74159$
  • $23-37-4159$
  • $2-337-4159$
  • $2-3-374159$
  • $233-74159$
  • $23-374159$
  • $2-3374159$

    $23374159$ is the smallest 8 digit number with exactly $12$ "breakdowns", I don't know whether there exists another $8$ digit number with exactly $12$ breakdowns.

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If that $8$ digit number is a prime itself and also counted as a breakdown, then the smallest $8$ digit numbers with exactly $12$ breakdowns are $23373613$, $23374159$, and $23379397$ :

  • $2-3-3-7-3613$
  • $2-3-3-73-613$
  • $2-3-3-73613$
  • $2-3-37-3613$
  • $2-3-373-613$
  • $2-3-373613$
  • $2-337-3613$
  • $23-37-3613$
  • $23-373-613$
  • $23-373613$
  • $233-73613$
  • $23373613$
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