## Twin prime conjecture.

Posted on: May 24th, 2013 by
1

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 $p_n$ be the sequence of prime numbers and $q_n=p_{n+1}-p_n$.
Exercise: Show that $\limsup_{n\rightarrow \infty} q_n = \infty$.
Now the twin prime conjecture states that $\liminf_{n\rightarrow \infty} q_n = 2$.
We don't even know if $\liminf_{n\rightarrow \infty} q_n < \infty$.
That's where the mathematician Yitang Zhang made progress. He showed that $\liminf_{n\rightarrow \infty} q_n \leq 70 000 000$.
Thus clearly the sequence $q_n$ does not have a limit.