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?