Lim sup of difference of primes

[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?

 

[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.

 

[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.

 

[Dragonfall]

Yes, that was a typo. I meant 0.

 

[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).