Here's an interesting prime number problem.
We know that there is no highest prime number (trivial). We have also proved that, as we go up the number line, prime numbers get scarcer (decidedly non-trivial) and that there is no highest distance between two consecutive prime numbers (a moderately obvious corrolary thereof).
What I want to know is: is it true that all even distances between consecutive prime numbers occur?
Obviously it's not true for all positive distances, because there's only one even prime, so odd distances must always involve 2, and the only prime number consecutive to 2 is 3. Odd distances other than 1 are impossible. But it might be true for even distances.
Next question: if the above is not true, is it true that al even distances occur between prime numbers, not necessarily consecutive ones?
(This one is definitely not true for odd distances, either: 2+7 isn't prime.)
For the record: I have no idea whether either conjecture is true. I don't even know where to start solving it. But someone on this forum might. And if not, someone on this forum might win a Fields Medal for solving them!