Recently, a chinese mathematician made progress towards the twin prime conjecture. For an informal introduction, read it in the news here.
I cannot resist being formal, so let me give a puzzle and explain the conjecture as well:
Let be the sequence of prime numbers and .
Exercise: Show that .
Now the twin prime conjecture states that .
We don't even know if .
That's where the mathematician Yitang Zhang made progress. He showed that .
Thus clearly the sequence does not have a limit.