Gaps between prime-roots

[cloitre]

le 2/02/03 10:46, Rainer Rosenthal à r.rosenthal at web.de a écrit :

> From: Neil Fernandez
>> Rainer Rosenthal
>>> In the course of thread "Primes..." in sci.math I proposed
>>> the following:
>>> 
>>> 1, 113, 1327, ... ???             (1)
>>> 
>>> where a(n) = smallest natural such that
>>> sqrt(P_(k+1)) - sqrt(P_k) < 1/n  for all k with P_k > a(n)
>> 
>> Are the terms 113 and 1327 proven?
>> 
> 
> No.
> 
> That's why I asked here instead of throwing in some nonsense
> into the OEIS.
> 
> I like your next sequence
> 2,2,2,17,29,41,59,71,101,101,137,149,179,197,227,269,...
> 
> but the thrill with (1) above is: can it really be found?
> I didn't even succeed in proving that
> 
> sqrt(P_(k+1)) - sqrt(P_k) < 1/2
> 
> must hold for all k beyond some K, let alone that with P_K=113.
> Even the famous Bertrand postulate (a theorem) gives me only
> 
> sqrt(P_(k+1)) - sqrt(P_k) ~< sqrt(P_k)/2
> 
> or as I prefer writing:  w'-w ~< w/2, where w and w' are the
> square roots of prime p and successor-prime p'.
> Proof:  (w'+w)(w'-w)=p'-p < 2p, so w'-w < p/(w'+w) ~ (p/(2w)=w/2.
> 
> (Pun intended: p-prime is not prime p but the next one).
> 
> Rainer Rosenthal
> r.rosenthal at web.de
> 
> 
> 

You are dealing somewhat with Andrica's conjecture :

It is even not proved that :

sqrt(P_(k+1)) - sqrt(P_k) < 1


http://mathworld.wolfram.com/AndricasConjecture.html

Neil, there is also a duplicate :

A074976 = A078692 sorry



Benoit Cloitre