# Maximum number of breakdowns for an $8$ digit number

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?

• Simply write a program to check it out if it is of great importance to you. – principal-ideal-domain Aug 21 '15 at 19:53
• @principal-ideal-domain, thanks, but I'm not a computer programmer at all – Level- 5c Being Aug 21 '15 at 19:58
• 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. – principal-ideal-domain Aug 21 '15 at 20:03
• @principal-ideal-domain, I conjecture that the maximum would not exceed 18 – Level- 5c Being Aug 21 '15 at 20:16
• I am quite sure that the first 4 digits of that number must be 2337 – Level- 5c Being Aug 21 '15 at 21:34

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.

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 $23379397$ :

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