i.e. problem statements that someone who has a degree but is not a mathematician would still be able to appreciate. I'm really interested in this, but reading through the list it all sounds really "deep" =)
I was wondering what was the most surprising problem to not have been solved yet.
1. Goldbach's conjecture: Can every even integer greater than 2 be written as the sum of two primes?
2. Twin prime conjecture: Are there infinitely many primes p such that p + 2 is prime?
3. Legendre's conjecture: Does there always exist at least one prime between consecutive perfect squares?
4. Are there infinitely many primes p such that p − 1 is a perfect square? In other words: Are there infinitely many primes of the form n^2 + 1?
I don't think any of them has a million dollar prize, but tenure at a decent university seems like a fairly reasonable expectation for solving one of these.
No "small" solutions exist, thanks to exhaustive search, but the prospect that one does exist at a horrifyingly large number remains.