# Lim sup of difference of primes

1. Aug 10, 2006

### Dragonfall

Although Andrica's conjecture is still unsolved, I'm told that it is possible to prove that

$$\lim\sup_{n\rightarrow\infty}\sqrt{p_{n+1}}-\sqrt{p_n}=1$$.

Does anyone know how or can point me to a source?

2. Aug 10, 2006

### shmoe

Who told you that was true? It looks an awful lot like the max occurs when n=4, so the primes 7 and 11 and seems to decrease from there. I've seen it conjectured that the full limit is actually zero, not much of a conjecture if the lim sup was known to be 1.

3. Aug 10, 2006

### CRGreathouse

I can't imagine the limit being higher than 0. Heck, find a number that makes it go higher than 0.01 for n > 10^9 and I'll be suprised.

4. Aug 12, 2006

### Dragonfall

Yes, that was a typo. I meant 0.

5. Aug 12, 2006

### shmoe

If the lim sup was 0, then the limit would be 0. This was still an unsolved problem according to Guy's 2004 "unsolved problems in number theory".

Maybe they meant

$$\lim\inf_{n\rightarrow\infty}\sqrt{p_{n+1}}-\sqrt{p_n}=0$$

which you can manage. Use the fact that $$p_{n+1}-p_{n}\leq \log p_n$$ is true infinitely often (much more is true actually, see Goldstom, Pintz and Yildrims recent work).