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?