My son Horatio (nine years old, fourth grade) came home with some fun math homework exercises today. One of his problems was the following little question: 


  I am thinking of a number...
  
  
  It is prime.
  The digits add up to $10.$ 
  It has a $3$ in the tens place.
  
  
  What is my number? 


Let us assume that the problem refers to digits in decimal notation. Horatio came up with $37,$ of course, and asked me whether there might be larger solutions with more digits. We observed together that $433$ is another solution, and also $631$ and $1531.$ But also notice that $10333$ solves the problem, based on the list of the first $10000$ primes, and also $100333$, and presumably many others. 

My question is: How many solutions does the problem have? In particular, are there infinitely many solutions? 

How could one prove or refute such a thing? I could imagine that there are very large prime numbers of the decimal form $10000000000000\cdots00000333$, but don't know how to prove or refute this.

Can you provide a satisfactory answer this fourth-grade homework question? 

As requested I'm posting this an answer. I wrote a short sage script to check the primality of numbers of the form $10^n+333$ where $n$ is in the range $[4,2000]$. I found that the following values of $n$ give rise to prime numbers:

$$4,5,6,12,53,222,231,416.$$

Edit 3: I stopped my laptop's search between 2000 and 3000, since it hadn't found anything in 20 minutes. I wrote a quick program to check numbers of the form $10^n+3*10^i+33$. Here are a couple 


100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000030000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000033
100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000300033
100000000000000000000000000000000000000000000000000000300000000000000000000000000000000000000033
100000000000000000000000000000000000000000000000030000000000000000000000000000000000000000000033
100000000000000000000000000000000000000000000030000000000000000000000000000000000000000000000033
10000000000000000000000000000000003000000033
10000000000000000000000000000030000000000033
10000000000000000000000030000000000000000033
10000000003000000000000000000000000000000033


There seemed to be plenty of numbers of this form and presumably I could find more if I checked some of the other possible forms as outlined by dr jimbob. 

Note: I revised the post a bit after jimbob pointed out I was actually looking for primes that didn't quite fit the requirements. 

Edit 4: As requested here are the sage scripts I used. 
To check if $10^n+333$ was prime:

for n in range(0,500):
  k=10^n+333
  if(is_prime(k)):
    print n


And to check for numbers of the form $10^n+3*10^i+33$:

for n in range(0,500):
  k=10^n+33
  for i in range(2,n):
    l=k+3*10^i
    if(is_prime(l)):
      print l

