Shavitian subprime numbers, which may also be known by another term if anyone else has bothered to care about such numbers, are interesting things! Here's the formal definition:
A positive integer N is a Shavitian subprime if any subset of consecutive digits within N, including N itself, is prime.
For instance, for the number 1,234 to be a Shavitian subprime, the following numbers would all have to be prime:




It turns out there are only 9 such numbers. They are:



It's actually possible to prove that these are the only Shavitian subprimes for all n > 1.
Proof of Shavitian subprimes
The trick to proving that there are no other Shavitian subprimes lies in how we can create candidates for subprimes. These candidates are not necessarily subprimes, but any multidigit subprime can be created by this method. In other words, not every candidate is a subprime, but every subprime is a candidate. To create a candidate, take a known Shavitian subprime and append 2, 3, 5 or 7 to the left of it.
If we have all Ndigit Shavitian subprimes, we can create a list of candidates that would contain all (N+1)digit Shavitian subprimes by appending 2, 3, 5 and 7 to each Ndigit number. Appending any other digit would introduce a nonprime digit to the number, thus making it not a Shavitian subprime. And we can prove that every (N+1)digit Shavitian subprime is in the list of candidates^{1}.
It's easy to create a list of all possible 1digit
Shavitian subprimes; they're just all onedigit prime numbers. We can then
use these to create all 2digit candidates, and use those to
create all 3digit candidates, etc. This creates a tree structure, with each number spawning four "children" derived from the above method. At each step, before we create the list of M_{14} candidates from a candidate N, we check to see if N is actually a Shavitian subprime; if it isn't, none of the Ms or their children will be, so we can abandon that branch of the tree. (Excuse the mixed metaphors; I didn't come up with these terms.)
Below is the list. As you can see, all of the branches end fairly quickly. Of all the 16 2digit candidates, only 4 are actually Shavitian subprimes. And of the 16 candidates derived from those, only 1 is a Shavitian subprime. And none of its children are Shavitian subprimes. Remember, there can't be any Shavitian subprime that is not in this tree, as we showed above (and proved in the footnote).
(Note: In this list, a black number is a Shavitian subprime, while a red number is not. Also, at times I have notation x2 or x5. Any number that has a digit followed by 2 or 5 is not a Shavitian subprime, since that x2 or x5 subset would be divisible by 2 or 5. Each red number is followed by an ellipsis to signify that there are actually infinite candidates derivable from it, but none of them will be Shavitian subprimes, since they'd all contain that number.)
 2
 x2...
 3
 23
 x23...
 33... (divisible by 3)
 53
 x53...
 73
 273... (divisible by 3)
 373
 2373... (divisible by 3)
 3373... (33 is divisible by 3)
 5373... (divisible by 3)
 7373... (divisible by 73)
 573... (divisible by 3)
 773... (77 is divisible by 7)
 5
 x5...
 7
 27... (divisible by 3. Incidentally, 27 is my favorite number.)
 37
 237... (divisible by 3)
 337... (33 is divisible by 3)
 537... (divisible by 3)
 737... (divisible by 11)
 57... (divisible by 3)
 77... (divisible by 7)