I am sitting with two other people who are wearing hats with the prime numbers 5 and 11. My hat has a prime number also, but I don't know what it is, nor do each of them know their numbers. It is also known that our three numbers represent side lengths forming a triangle having a perimeter length which is also a prime number. They are each asked if they know their number, and each says no. I answer no also. They are asked again, and say no, and now it is my turn. What is my number?
Note: I have figured out that the only possible numbers I can have are 7 and 13, but I can't seem to narrow it down to which is my number.