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Twin prime conjecture.

Posted on: May 24th, 2013 by

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.

One Response to Twin prime conjecture.

  1. Mathav had this to say about that:

    n!+2 , n!+3, n!+4, ... , n!+n are all composite for n>=2. This shows that lim sup q_n is infinity.

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