13. To steal a precious jewel, a thief must discover the code that allows him to open the safe's door. The information he has managed to gather is as follows:

- the code is a number
- any consecutive subsequence of digits of the code (thus each digit taken individually, as well as each pair of digits, etc., up to the entire number) represents a prime number (for example, 217 is not good, because 1 is not a prime and 21 is not a prime)
- the code is the largest number that has this property.

What is the secret code to open the safe?
13. The answer is 373.

Let's find all the numbers that satisfy the conditions of the text.

First, we notice that if we have found all acceptable numbers of $n$ digits, then an acceptable number of $n+1$ digits must necessarily contain one of the numbers of $n$ digits. In particular, if we discover that there is no number of $k$ digits, then there will be no acceptable number with more than $k$ digits.

The acceptable numbers of one digit are the one-digit primes, i.e., $2, 3, 5,$ and $7$. We note, however, that the digits 2 and 5 can only be at the beginning of an acceptable number (otherwise, the number would have a subsequence divisible by 2 or 5, respectively).

The acceptable numbers of two digits are therefore $23, 37, 53,$ and $73$.

Now let's look at the numbers of three digits. None can end in 23 or 53. Then 237, 537, 273, and 573 are not acceptable as they are multiples of 3. No number can start with the same pair of digits, otherwise it would have a subsequence that is a multiple of 11 (for example, in 773, the subsequence 77 is a multiple of 11). The only ones left to check are 373, which turns out to be prime (and therefore acceptable), and 737, which is a multiple of 11.

Now let's see if there can be numbers of four digits. We only need to check 4: 2373, 3373, 5373, and 7373. The first and the third contain a subsequence that is a multiple of 3. The second contains a subsequence that is a multiple of 11, and the last one is a multiple of 101. Therefore, there are no acceptable numbers of four digits, and, as stated at the beginning, there will be no acceptable numbers with more than four digits. The code we are looking for is thus 373.